MATLAB® Test Report

Timestamp:

12-Jan-2026 16:12:02

Host:

runnervmi13qx

Platform:

glnxa64

MATLAB Version:

25.1.0.2973910 (R2025a) Update 1

Number of Tests:

21

Testing Time:

90.4814 seconds

Overall Result:

PASSED

Overview

/home/runner/work/Applied-ODEs/Applied-ODEs/SoftwareTests/

SmokeTests

41.6283 seconds

SolnSmokeTests

48.8531 seconds

Details

/home/runner/work/Applied-ODEs/Applied-ODEs/SoftwareTests/

SmokeTests

SmokeRun Class Setup Parameters: Project=matlab.project.Project Test Parameters: File=CharacteristicEquations.mlx

The test passed. Duration: 22.1278 seconds

Events:

Timestamp: 12-Jan-2026 16:10:40

Verbosity: Terse

Logged Diagnostic:

Figure saved to: --> /tmp/d6eeaa4f-d848-4515-a428-a5c6d6fe05f2/Figure_30c91009-ef45-4a6e-a8c1-fb32c60ddb13.png

Event Location: SmokeTests[Project=matlab.project.Project]/SmokeRun(File=CharacteristicEquations.mlx)

Stack:

In /home/runner/work/Applied-ODEs/Applied-ODEs/SoftwareTests/SmokeTests.m (SmokeTests.SmokeRun) at 94

Timestamp: 12-Jan-2026 16:10:47

Verbosity: Terse

Logged Diagnostic:

Figure saved to: --> /tmp/d6eeaa4f-d848-4515-a428-a5c6d6fe05f2/Figure_7a6df21b-3f1b-4d06-8d3e-e506f8aeee3e.png

Event Location: SmokeTests[Project=matlab.project.Project]/SmokeRun(File=CharacteristicEquations.mlx)

Stack:

In /home/runner/work/Applied-ODEs/Applied-ODEs/SoftwareTests/SmokeTests.m (SmokeTests.SmokeRun) at 94

(Overview)

SmokeRun Class Setup Parameters: Project=matlab.project.Project Test Parameters: File=CheckYourWork.mlx

The test passed. Duration: 0.1885 seconds

(Overview)

SmokeRun Class Setup Parameters: Project=matlab.project.Project Test Parameters: File=Classification.mlx

The test passed. Duration: 0.0981 seconds

(Overview)

SmokeRun Class Setup Parameters: Project=matlab.project.Project Test Parameters: File=IntegratingFactors.mlx

The test passed. Duration: 4.8831 seconds

Event:

Timestamp: 12-Jan-2026 16:10:51

Verbosity: Terse

Logged Diagnostic:

Figure saved to: --> /tmp/d6eeaa4f-d848-4515-a428-a5c6d6fe05f2/Figure_8f751484-a61c-4331-8180-69ffea87df77.png

Event Location: SmokeTests[Project=matlab.project.Project]/SmokeRun(File=IntegratingFactors.mlx)

Stack:

In /home/runner/work/Applied-ODEs/Applied-ODEs/SoftwareTests/SmokeTests.m (SmokeTests.SmokeRun) at 94

(Overview)

SmokeRun Class Setup Parameters: Project=matlab.project.Project Test Parameters: File=SeparationOfVariables.mlx

The test passed. Duration: 3.6274 seconds

Event:

Timestamp: 12-Jan-2026 16:10:54

Verbosity: Terse

Logged Diagnostic:

Figure saved to: --> /tmp/d6eeaa4f-d848-4515-a428-a5c6d6fe05f2/Figure_afdb3490-dbd9-4867-9a1e-e8e4cbd89bb7.png

Event Location: SmokeTests[Project=matlab.project.Project]/SmokeRun(File=SeparationOfVariables.mlx)

Stack:

In /home/runner/work/Applied-ODEs/Applied-ODEs/SoftwareTests/SmokeTests.m (SmokeTests.SmokeRun) at 94

(Overview)

SmokeRun Class Setup Parameters: Project=matlab.project.Project Test Parameters: File=SystemsOfODEs.mlx

The test passed. Duration: 4.0304 seconds

Event:

Timestamp: 12-Jan-2026 16:10:58

Verbosity: Terse

Logged Diagnostic:

Figure saved to: --> /tmp/d6eeaa4f-d848-4515-a428-a5c6d6fe05f2/Figure_4c3d5c70-dcfa-4927-b899-3f62dadf6950.png

Event Location: SmokeTests[Project=matlab.project.Project]/SmokeRun(File=SystemsOfODEs.mlx)

Stack:

In /home/runner/work/Applied-ODEs/Applied-ODEs/SoftwareTests/SmokeTests.m (SmokeTests.SmokeRun) at 94

(Overview)

SmokeRun Class Setup Parameters: Project=matlab.project.Project Test Parameters: File=UndeterminedCoefficients.mlx

The test passed. Duration: 6.6730 seconds

Events:

Timestamp: 12-Jan-2026 16:11:03

Verbosity: Terse

Logged Diagnostic:

Figure saved to: --> /tmp/d6eeaa4f-d848-4515-a428-a5c6d6fe05f2/Figure_3f28787f-adb2-4bfd-b69a-18172673ba50.png

Event Location: SmokeTests[Project=matlab.project.Project]/SmokeRun(File=UndeterminedCoefficients.mlx)

Stack:

In /home/runner/work/Applied-ODEs/Applied-ODEs/SoftwareTests/SmokeTests.m (SmokeTests.SmokeRun) at 94

Timestamp: 12-Jan-2026 16:11:07

Verbosity: Terse

Logged Diagnostic:

Figure saved to: --> /tmp/d6eeaa4f-d848-4515-a428-a5c6d6fe05f2/Figure_1bf8425b-461b-4414-bd2a-b38e79d5d72b.png

Event Location: SmokeTests[Project=matlab.project.Project]/SmokeRun(File=UndeterminedCoefficients.mlx)

Stack:

In /home/runner/work/Applied-ODEs/Applied-ODEs/SoftwareTests/SmokeTests.m (SmokeTests.SmokeRun) at 94

(Overview)

SolnSmokeTests

ExistSolns Class Setup Parameters: Project=matlab.project.Project Test Parameters: File=CharacteristicEquations.mlx

The test passed. Duration: 0.0657 seconds

(Overview)

ExistSolns Class Setup Parameters: Project=matlab.project.Project Test Parameters: File=CheckYourWork.mlx

The test passed. Duration: 0.0057 seconds

(Overview)

ExistSolns Class Setup Parameters: Project=matlab.project.Project Test Parameters: File=Classification.mlx

The test passed. Duration: 0.0054 seconds

(Overview)

ExistSolns Class Setup Parameters: Project=matlab.project.Project Test Parameters: File=IntegratingFactors.mlx

The test passed. Duration: 0.0055 seconds

(Overview)

ExistSolns Class Setup Parameters: Project=matlab.project.Project Test Parameters: File=SeparationOfVariables.mlx

The test passed. Duration: 0.0075 seconds

(Overview)

ExistSolns Class Setup Parameters: Project=matlab.project.Project Test Parameters: File=SystemsOfODEs.mlx

The test passed. Duration: 0.0049 seconds

(Overview)

ExistSolns Class Setup Parameters: Project=matlab.project.Project Test Parameters: File=UndeterminedCoefficients.mlx

The test passed. Duration: 0.0051 seconds

(Overview)

SmokeRun Class Setup Parameters: Project=matlab.project.Project Test Parameters: File=CharacteristicEquations.mlx

The test passed. Duration: 11.4992 seconds

Events:

Timestamp: 12-Jan-2026 16:11:15

Verbosity: Terse

Logged Diagnostic:

Figure saved to: --> /tmp/d6eeaa4f-d848-4515-a428-a5c6d6fe05f2/Figure_dae85f87-d416-4731-91c4-f887b386645f.png

Event Location: SolnSmokeTests[Project=matlab.project.Project]/SmokeRun(File=CharacteristicEquations.mlx)

Stack:

In /home/runner/work/Applied-ODEs/Applied-ODEs/SoftwareTests/SolnSmokeTests.m (SolnSmokeTests.SmokeRun) at 110

Timestamp: 12-Jan-2026 16:11:19

Verbosity: Terse

Logged Diagnostic:

Figure saved to: --> /tmp/d6eeaa4f-d848-4515-a428-a5c6d6fe05f2/Figure_a8ebbc79-18a4-4c59-9777-5d783f149a68.png

Event Location: SolnSmokeTests[Project=matlab.project.Project]/SmokeRun(File=CharacteristicEquations.mlx)

Stack:

In /home/runner/work/Applied-ODEs/Applied-ODEs/SoftwareTests/SolnSmokeTests.m (SolnSmokeTests.SmokeRun) at 110

(Overview)

SmokeRun Class Setup Parameters: Project=matlab.project.Project Test Parameters: File=CheckYourWork.mlx

The test passed. Duration: 3.8486 seconds

Event:

Timestamp: 12-Jan-2026 16:11:21

Verbosity: Terse

Logged Diagnostic:

Figure saved to: --> /tmp/d6eeaa4f-d848-4515-a428-a5c6d6fe05f2/Figure_bcc227f9-220d-4450-af92-6d9eecfeaee4.png

Event Location: SolnSmokeTests[Project=matlab.project.Project]/SmokeRun(File=CheckYourWork.mlx)

Stack:

In /home/runner/work/Applied-ODEs/Applied-ODEs/SoftwareTests/SolnSmokeTests.m (SolnSmokeTests.SmokeRun) at 110

(Overview)

SmokeRun Class Setup Parameters: Project=matlab.project.Project Test Parameters: File=Classification.mlx

The test passed. Duration: 0.0801 seconds

(Overview)

SmokeRun Class Setup Parameters: Project=matlab.project.Project Test Parameters: File=IntegratingFactors.mlx

The test passed. Duration: 3.8619 seconds

Event:

Timestamp: 12-Jan-2026 16:11:24

Verbosity: Terse

Logged Diagnostic:

Figure saved to: --> /tmp/d6eeaa4f-d848-4515-a428-a5c6d6fe05f2/Figure_5d759ee1-3c14-48a1-9693-13eca8a1e9b7.png

Event Location: SolnSmokeTests[Project=matlab.project.Project]/SmokeRun(File=IntegratingFactors.mlx)

Stack:

In /home/runner/work/Applied-ODEs/Applied-ODEs/SoftwareTests/SolnSmokeTests.m (SolnSmokeTests.SmokeRun) at 110

(Overview)

SmokeRun Class Setup Parameters: Project=matlab.project.Project Test Parameters: File=SeparationOfVariables.mlx

The test passed. Duration: 3.8951 seconds

Event:

Timestamp: 12-Jan-2026 16:11:28

Verbosity: Terse

Logged Diagnostic:

Figure saved to: --> /tmp/d6eeaa4f-d848-4515-a428-a5c6d6fe05f2/Figure_f91fde89-e6f1-4798-a160-9201a215c531.png

Event Location: SolnSmokeTests[Project=matlab.project.Project]/SmokeRun(File=SeparationOfVariables.mlx)

Stack:

In /home/runner/work/Applied-ODEs/Applied-ODEs/SoftwareTests/SolnSmokeTests.m (SolnSmokeTests.SmokeRun) at 110

(Overview)

SmokeRun Class Setup Parameters: Project=matlab.project.Project Test Parameters: File=SystemsOfODEs.mlx

The test passed. Duration: 14.9901 seconds

Event:

Timestamp: 12-Jan-2026 16:11:46

Verbosity: Terse

Logged Diagnostic:

Figure saved to: --> /tmp/d6eeaa4f-d848-4515-a428-a5c6d6fe05f2/Figure_bb8aed83-706f-454f-ab30-221e4d1cbfed.png

Event Location: SolnSmokeTests[Project=matlab.project.Project]/SmokeRun(File=SystemsOfODEs.mlx)

Stack:

In /home/runner/work/Applied-ODEs/Applied-ODEs/SoftwareTests/SolnSmokeTests.m (SolnSmokeTests.SmokeRun) at 110

(Overview)

SmokeRun Class Setup Parameters: Project=matlab.project.Project Test Parameters: File=UndeterminedCoefficients.mlx

The test passed. Duration: 10.5783 seconds

Events:

Timestamp: 12-Jan-2026 16:11:52

Verbosity: Terse

Logged Diagnostic:

Figure saved to: --> /tmp/d6eeaa4f-d848-4515-a428-a5c6d6fe05f2/Figure_c4f16d1c-1f5a-4ee1-a4e9-a6d50ad4f52c.png

Event Location: SolnSmokeTests[Project=matlab.project.Project]/SmokeRun(File=UndeterminedCoefficients.mlx)

Stack:

In /home/runner/work/Applied-ODEs/Applied-ODEs/SoftwareTests/SolnSmokeTests.m (SolnSmokeTests.SmokeRun) at 110

Timestamp: 12-Jan-2026 16:11:56

Verbosity: Terse

Logged Diagnostic:

Figure saved to: --> /tmp/d6eeaa4f-d848-4515-a428-a5c6d6fe05f2/Figure_d080ade1-5aa2-450f-a944-00caa273b861.png

Event Location: SolnSmokeTests[Project=matlab.project.Project]/SmokeRun(File=UndeterminedCoefficients.mlx)

Stack:

In /home/runner/work/Applied-ODEs/Applied-ODEs/SoftwareTests/SolnSmokeTests.m (SolnSmokeTests.SmokeRun) at 110

(Overview)

Command Window Text

Running SmokeTests >> Running CharacteristicEquations.mlx Solve 12*y(x) - 7*diff(y(x), x) + diff(y(x), x, x) == 0 This is the default solution. The characteristic polynomial of a second order linear ODE will be quadratic. This is the default. Please compute and enter the roots of the characteristic polynomial. This is the default solution. Solve y(x) - 2*diff(y(x), x) + diff(y(x), x, x) == 0 This is the default solution. The characteristic polynomial of a second order linear ODE will be quadratic. This is the default. Please compute and enter the roots of the characteristic polynomial. This is the default solution. Solve 5*y(x) - diff(y(x), x) + 5*diff(y(x), x, x) == 0 This is the default solution. The characteristic polynomial of a second order linear ODE will be quadratic. This is the default. Please compute and enter the roots of the characteristic polynomial. This is the default solution. Solve 50*y(x) + 20*diff(y(x), x) + 2*diff(y(x), x, x) == 0 This is the default solution. The characteristic polynomial of a second order linear ODE will be quadratic. This is the default. Please compute and enter the roots of the characteristic polynomial. This is the default value. Once you generate a problem, there will certainly be known roots. This is the default solution. [Terse] Diagnostic logged (2026-01-12 16:10:40): Figure saved to: --> /tmp/d6eeaa4f-d848-4515-a428-a5c6d6fe05f2/Figure_30c91009-ef45-4a6e-a8c1-fb32c60ddb13.png [Terse] Diagnostic logged (2026-01-12 16:10:47): Figure saved to: --> /tmp/d6eeaa4f-d848-4515-a428-a5c6d6fe05f2/Figure_7a6df21b-3f1b-4d06-8d3e-e506f8aeee3e.png .>> Running CheckYourWork.mlx .>> Running Classification.mlx Please select one or more variables. Please select an answer. Please select one or more variables. Please select an answer. Please enter a nonzero order. Please select an answer. Please select an answer. Please enter a nonzero order. Please select an answer. Please select an answer. Please enter a nonzero order. Please select an answer. Please select an answer. .>> Running IntegratingFactors.mlx Please enter a nonzero expression. Please enter a nonzero expression. Please enter a nonzero expression. Please enter a nonzero expression. Please enter a nonzero expression. Please enter a nonzero expression. Please enter a nonzero expression. Please enter a nonzero expression. Please enter a nonzero expression. Please enter a nonzero expression. Please enter a nonzero expression. Please enter a positive integer. We start with our generic formula for chemical concentration: diff(M(t), t) == concentrationIn*rateIn - (rateOut*M(t))/(volInit + t*(rateIn - rateOut)) Plugging in the parameters, we have: diff(M(t), t) == 4 - M(t)/(50*(t/50 + 100)) [Terse] Diagnostic logged (2026-01-12 16:10:51): Figure saved to: --> /tmp/d6eeaa4f-d848-4515-a428-a5c6d6fe05f2/Figure_8f751484-a61c-4331-8180-69ffea87df77.png .>> Running SeparationOfVariables.mlx Please select an answer. Please select an answer. Please select an answer. Please enter a nonzero expression. Please enter a nonzero expression. Please enter a nonzero expression. Please enter a nonzero expression. Please enter a nonzero expression. Please enter a nonzero expression. Please enter a nonzero expression. Please enter a nonzero expression. Please enter a positive integer. Please enter a nonzero expression. Please enter a nonzero expression. Please enter a nonzero expression. P(t) == (P0*k*exp(r*t))/((k - P0) + P0*exp(r*t)) Plugging in these custom parameters, we get: P(t) == (86100000*exp(t/20))/(8200*exp(t/20) + 2300) Here's a plot of the solution: [Terse] Diagnostic logged (2026-01-12 16:10:54): Figure saved to: --> /tmp/d6eeaa4f-d848-4515-a428-a5c6d6fe05f2/Figure_afdb3490-dbd9-4867-9a1e-e8e4cbd89bb7.png .>> Running SystemsOfODEs.mlx Consider the following system of differential equations dx/dt == 3*x + y dy/dt == 3*x + 2*y This is the default. Please enter a non-zero matrix. Please enter the eigenvalues of A. Incorrect. Solve the equation (A - lambda*I)*v == 0 Incorrect. Solve the equation (A - lambda*I)*v == 0 Let beta == 0.2 and gamma == 0.15. Incorrect. In this case, A should be a 2x2 matrix. Please enter the eigenvalues of A. You can be more precise than this. Please select a different answer. Basic reproduction number R₀ = 2 R₀ > 1: Epidemic will occur This is the default. Please enter the correct matrix. This is the default. Please enter the computed eigenvalues. [Terse] Diagnostic logged (2026-01-12 16:10:58): Figure saved to: --> /tmp/d6eeaa4f-d848-4515-a428-a5c6d6fe05f2/Figure_4c3d5c70-dcfa-4927-b899-3f62dadf6950.png .>> Running UndeterminedCoefficients.mlx The proposed solution is y == (2*x)/9 + C2*exp(x) + x^2/3 + C1*exp(3*x) + 38/27. Plugging into the original ODE we have - 2*x + x^2 + 4 == - 2*x + x^2 + 4, which is true! Solve - 192*y(x) + 3*diff(y(x), x, x) == - 4*x - 5*x^2 - 2*x^3 - 5*x^4 + 4 This is the default. Please enter a solution and resubmit. This is the default. Please enter a solution and resubmit. This is the default. Please enter a solution and resubmit. This is the default, please enter a proposed solution. Solve 9*y(x) + 6*diff(y(x), x) + diff(y(x), x, x) == -2*exp(2*x) This is the default. Please enter a solution and resubmit. This is the default. Please enter a solution and resubmit. ans = logical 0 This is the default. Please enter a solution and resubmit. ans = logical 0 Solve 5*y(x) - 3*diff(y(x), x) + diff(y(x), x, x) == 2*cos(4*x) This is the default. Please enter a solution and resubmit. This is the default. Please enter a solution and resubmit. ans = logical 0 This is the default. Please enter a solution and resubmit. ans = logical 0 Solve 17*y(x) - 8*diff(y(x), x) + diff(y(x), x, x) == -2*cos(4*x)*exp(x) This is the default. Please enter a solution and resubmit. This is the default. Please enter a solution and resubmit. ans = logical 0 This is the default. Please enter a solution and resubmit. ans = logical 0 This is the default value. The solution is not just the forcing function. Parameters: γ=1.00, ω₀=0.50, F₀=0.50, ω=0.10 -(exp(-t/2)*(7*t - 2))/2 This is the default. The spring constant is not 0. Bond wavenumber: 2886, External wavenumber: 2880, Response Amplitude: 8.1353e-29 [Terse] Diagnostic logged (2026-01-12 16:11:03): Figure saved to: --> /tmp/d6eeaa4f-d848-4515-a428-a5c6d6fe05f2/Figure_3f28787f-adb2-4bfd-b69a-18172673ba50.png [Terse] Diagnostic logged (2026-01-12 16:11:07): Figure saved to: --> /tmp/d6eeaa4f-d848-4515-a428-a5c6d6fe05f2/Figure_1bf8425b-461b-4414-bd2a-b38e79d5d72b.png . Done SmokeTests __________ Running SolnSmokeTests .......>> Running CharacteristicEquationsSoln.mlx Solve 15*y(x) - 8*diff(y(x), x) + diff(y(x), x, x) == 0 Correct! The characteristic polynomial is: - 8*x + x^2 + 15 == 0 Correct! The roots are r_1 == 3, and r_2 == 5. Correct! The solution is: y(x) == C1*exp(3*x) + C2*exp(5*x) Solve diff(y(x), x, x) == 0 Correct! The characteristic polynomial is: x^2 == 0 Correct! The solution is r == 0 twice! Correct! The solution is: y(x) == C2 + C1*x Solve - 2*y(x) + 3*diff(y(x), x) - 4*diff(y(x), x, x) == 0 Correct! The characteristic polynomial is: 3*x - 4*x^2 - 2 == 0 Correct! The roots are r_1 == - (23^(1/2)*1i)/8 + 3/8, and r_2 == (23^(1/2)*1i)/8 + 3/8. Correct! The solution is: y(x) == C1*exp((3*x)/8)*cos((23^(1/2)*x)/8) - C2*exp((3*x)/8)*sin((23^(1/2)*x)/8) Solve 27*y(x) + 18*diff(y(x), x) + 3*diff(y(x), x, x) == 0 Correct! The characteristic polynomial is: 18*x + 3*x^2 + 27 == 0 Correct! The solution is r == -3 twice! Correct, this problem has repeated roots. Correct! The solution is: y(x) == C1*exp(-3*x) + C2*x*exp(-3*x) --- RLC Circuit Analysis --- L = 3.50 H, R = 0.50 kΩ = 500 Ω , C = 0.500 µF = 5.00 * 1e-7 F Characteristic equation: r² + 142.857r + 571428.571 = 0 Damping: 142.857 Natural frequency: 1511.858 Discriminant: -2265306.122 System type: Underdamped Complex roots: r = -71.429 ± 752.547i Assume q(0)=1 and q'(0)=0. Differential equation solution: q(t) = (exp(-(500*t)/7)*(79456894976000007*cos((3*52971263317333338^(1/2)*t)/917504) + 32768000*52971263317333338^(1/2)*sin((3*52971263317333338^(1/2)*t)/917504)))/79456894976000007 --- Mass-Spring-Damper Analysis --- m = 75.00 kg, c = 38.00 N·s/m, k = 521.00 N/m Initial conditions: x₀ = -1.00 m, v₀ = 0.00 m/s Natural frequency: ωₙ = 2.636 rad/s Damping ratio: ζ = 0.096 Discriminant: -27.530 System type: Underdamped Complex roots: r = -0.253 ± 2.623i Damped frequency: ωd = 2.623 rad/s Differential equation solution: x(t) = -(exp(-(19*t)/75)*(38714*cos((38714^(1/2)*t)/75) + 19*38714^(1/2)*sin((38714^(1/2)*t)/75)))/38714 [Terse] Diagnostic logged (2026-01-12 16:11:15): Figure saved to: --> /tmp/d6eeaa4f-d848-4515-a428-a5c6d6fe05f2/Figure_dae85f87-d416-4731-91c4-f887b386645f.png [Terse] Diagnostic logged (2026-01-12 16:11:19): Figure saved to: --> /tmp/d6eeaa4f-d848-4515-a428-a5c6d6fe05f2/Figure_a8ebbc79-18a4-4c59-9777-5d783f149a68.png .>> Running CheckYourWorkSoln.mlx MyGeneralSolution = (35*t)/18 + (7*t^2)/6 + C1*exp(2*t) + C2*exp(3*t) + 97/108 MySolution = (35*t)/18 + (37*exp(2*t))/4 - (193*exp(3*t))/27 + (7*t^2)/6 + 97/108 MyUnexpectedSolution = C1*airy(0, t) + C2*airy(2, t) MyNonSolution = C1 + int(exp(-t^3), t, 'IgnoreSpecialCases', true, 'IgnoreAnalyticConstraints', true) QuadraticEquation = a*x^2 + b*x + c == 0 Roots = -(b + (b^2 - 4*a*c)^(1/2))/(2*a) -(b - (b^2 - 4*a*c)^(1/2))/(2*a) My differential equation is: 6*x(t) - 5*diff(x(t), t) + diff(x(t), t, t) == 7*t^2 - 2 and the general solution is: (35*t)/18 + (37*exp(2*t))/4 - (193*exp(3*t))/27 + (7*t^2)/6 + 97/108 [Terse] Diagnostic logged (2026-01-12 16:11:21): Figure saved to: --> /tmp/d6eeaa4f-d848-4515-a428-a5c6d6fe05f2/Figure_bcc227f9-220d-4450-af92-6d9eecfeaee4.png .>> Running ClassificationSoln.mlx Correct! b and c are the independent variables in this equation. Correct! This is a partial differential equation, because it has more than one independent variable. Correct! b is the only independent variable in this equation. Correct! This is an ordinary differential equation, because it has one independent variable. Correct! The fourth derivative of y is the highest derivative that appears in the equation, so the order is 4. Correct! All coefficients of y and its derivatatives are constant or functions of t, so the equation is linear. Correct! The trivial solution y=0 is not a solution, so the equation is nonhomogeneous. Correct! The third derivative of y is the highest derivative that appears in the equation, so the order is 3. Correct! The equation has a nonlinear term y' * y, so the equation is nonlinear. Correct! Because the equation is nonlinear, we can't assess its homogeneity. Correct! The second derivative of y is the highest derivative that appears in the equation, so the order is 2. Correct! All coefficients of y and its derivatatives are constant or functions of t, so the equation is linear. Correct! The trivial solution y=0 is a solution, so the equation is homogeneous. . >> Running IntegratingFactorsSoln.mlx Correct! See below for a detailed explanation. Correct! See below for a detailed explanation. Correct! See below for a detailed explanation. Correct! See below for a detailed explanation. Correct! See below for a detailed explanation. Correct! See below for a detailed explanation. Correct! See below for a detailed explanation. Correct! See below for a detailed explanation. Correct! See below for a detailed explanation. Correct! See below for a detailed explanation. Correct! See below for a detailed explanation. Starting with the equation from part (c): M == (20000*t + 2*t^2 + C)/(5000 + t) Remember that M represents the *quantity* of mercury in the water, so we have M(0) = 1.5 * 100 = 150. Solve for C, plugging in M = 150 and t = 0: 150 == (0 + 0 + C)/(5000 + 0) 150 == C/5000 C == 750000 Finally, plugging this value for C back into the equation from part (c), we get: M == (20000*t + 2*t^2 + 750000)/(5000 + t) Correct! See below for a detailed explanation. We start with our generic formula for chemical concentration: diff(M(t), t) == concentrationIn*rateIn - (rateOut*M(t))/(volInit + t*(rateIn - rateOut)) Plugging in the parameters, we have: diff(M(t), t) == 4 - M(t)/(50*(t/50 + 100)) [Terse] Diagnostic logged (2026-01-12 16:11:24): Figure saved to: --> /tmp/d6eeaa4f-d848-4515-a428-a5c6d6fe05f2/Figure_5d759ee1-3c14-48a1-9693-13eca8a1e9b7.png .>> Running SeparationOfVariablesSoln.mlx Correct! This can be rewritten as: (1/y)*dy == 3*x^2*dx Correct! Using y' = dy/dx, this can be rewritten as: (1/y^2)*dy == (e^x/x)*dx Correct! There is no way to separate the x and y terms in this equation, so it is not separable. Correct! See below for a detailed explanation. Correct! See below for a detailed explanation. Correct! See below for a detailed explanation. Correct! See below for a detailed explanation. Correct! See below for a detailed explanation. Correct! See below for a detailed explanation. Correct! See below for a detailed explanation. Correct! See below for a detailed explanation. Correct! See below for a detailed explanation. Correct! See below for a detailed explanation. Correct! See below for a detailed explanation. Correct! See below for a detailed explanation. P(t) == (P0*k*exp(r*t))/((k - P0) + P0*exp(r*t)) Plugging in these custom parameters, we get: P(t) == (86100000*exp(t/20))/(8200*exp(t/20) + 2300) Here's a plot of the solution: [Terse] Diagnostic logged (2026-01-12 16:11:28): Figure saved to: --> /tmp/d6eeaa4f-d848-4515-a428-a5c6d6fe05f2/Figure_f91fde89-e6f1-4798-a160-9201a215c531.png .>> Running SystemsOfODEsSoln.mlx Consider the following system of differential equations dx/dt == - x - 3*y dy/dt == - 5*x + 5*y Correct! The system dx/dt == - x - 3*y dy/dt == - 5*x + 5*y can be written as: matrix([[dx/dt], [dy/dt]]) == matrix([[-1, -3], [-5, 5]])*matrix([[x], [y]]) So A == matrix([[-1, -3], [-5, 5]]) Correct! To find eigenvalues, solve det(A - lambda*I) == 0: det(matrix([[- lambda - 1, -3], [-5, - lambda + 5]])) == 0 - 4*lambda + lambda^2 - 20 == 0 lambda_1 == (4 + sqrt((-4)^2 - (-80)))/2, lambda_1 == 2*6^(1/2) + 2 lambda_2 == (4 - sqrt((-4)^2 - (-80)))/2, lambda_2 == - 2*6^(1/2) + 2 For lambda == 2*6^(1/2) + 2, solve (A - lambda*I)*v == 0 matrix([[- 2*6^(1/2) - 3, -3], [-5, - 2*6^(1/2) + 3]])*matrix([[v_1], [v_2]]) == matrix([[0], [0]]) matrix([[v_1], [v_2]]) == matrix([[- (2*6^(1/2))/5 + 3/5], [1]]) For lambda == - 2*6^(1/2) + 2, solve (A - lambda*I)*v == 0 matrix([[2*6^(1/2) - 3, -3], [-5, 2*6^(1/2) + 3]])*matrix([[v_1], [v_2]]) == matrix([[0], [0]]) matrix([[v_1], [v_2]]) == matrix([[(2*6^(1/2))/5 + 3/5], [1]]) x(t) == c_1*exp(-t*(2*6^(1/2) - 2))*((2*6^(1/2))/5 + 3/5) - c_2*exp(t*(2*6^(1/2) + 2))*((2*6^(1/2))/5 - 3/5) y(t) == c_1*exp(-t*(2*6^(1/2) - 2)) + c_2*exp(t*(2*6^(1/2) + 2)) Checking... matrix([[- c_1*exp(-t*(2*6^(1/2) - 2))*(2*6^(1/2) - 2)*((2*6^(1/2))/5 + 3/5) - c_2*exp(t*(2*6^(1/2) + 2))*(2*6^(1/2) + 2)*((2*6^(1/2))/5 - 3/5)], [c_2*exp(t*(2*6^(1/2) + 2))*(2*6^(1/2) + 2) - c_1*exp(-2*t*(6^(1/2) - 1))*(2*6^(1/2) - 2)]]) - matrix([[-1, -3], [-5, 5]])*matrix([[c_1*exp(-t*(2*6^(1/2) - 2))*((2*6^(1/2))/5 + 3/5) - c_2*exp(t*(2*6^(1/2) + 2))*((2*6^(1/2))/5 - 3/5)], [c_1*exp(-t*(2*6^(1/2) - 2)) + c_2*exp(t*(2*6^(1/2) + 2))]]) == matrix([[0], [0]]) Let beta == 0.2 and gamma == 0.17. Plugging in the values for beta == 0.2 and gamma == 0.17 we see that we get: Correct! The system dS/dt == -0.2*I dI/dt == 0.03*I which can be written as matrix([[dS/dt], [dI/dt]]) == matrix([[0, -0.2], [0, 0.03]])*matrix([[S], [I]]) So A == matrix([[0, -0.2], [0, 0.03]]) Correct! To find eigenvalues, solve det(A - lambda*I) == 0: det(matrix([[-1.0*lambda, -0.2], [0, - 1.0*lambda + 0.03]])) == 0 (lambda*(100*lambda - 3))/100 == 0 lambda_1 == (0.03 + sqrt((-0.03)^2 - 0.0))/2.0, lambda_1 == 0.03 lambda_2 == (0.03 - sqrt((-0.03)^2 - 0.0))/2.0, lambda_2 == 0.0 Not quite. What do positive eigenvalues mean for the long-term growth of solutions? Basic reproduction number R₀ = 1 The disease is endemic in the population. Correct! The coefficient matrix represents the building dynamics. A == matrix([[0, 1, 0, 0], [-2*omega^2, 0, omega^2, 0], [0, 0, 0, 1], [omega^2, 0, -omega^2, 0]]) dx_1/dt == v_1 dv_1/dt == - 2*omega^2*x_1 + omega^2*x_2 dx_2/dt == v_2 dv_2/dt == omega^2*x_1 - omega^2*x_2 Correct! The eigenvalues are ±(- 5^(1/2)/2 - 3/2)^(1/2) and ±(5^(1/2)/2 - 3/2)^(1/2), or approximately ±1.618i and ±0.618i, representing the natural frequencies of the building modes. The characteristic equation is: lambda^4 + 3*lambda^2 + 1 == 0 [Terse] Diagnostic logged (2026-01-12 16:11:46): Figure saved to: --> /tmp/d6eeaa4f-d848-4515-a428-a5c6d6fe05f2/Figure_bb8aed83-706f-454f-ab30-221e4d1cbfed.png .>> Running UndeterminedCoefficientsSoln.mlx The proposed solution is y == (2*x)/9 + C2*exp(x) + x^2/3 + C1*exp(3*x) + 38/27. Plugging into the original ODE we have - 2*x + x^2 + 4 == - 2*x + x^2 + 4, which is true! Solve - 5*y(x) + 4*diff(y(x), x) - diff(y(x), x, x) == - x - 5*x^2 + 5*x^3 + x^4 - 4 WorkOutGuess = A*x^4 + B*x^3 + C*x^2 + D*x + E Correct! The guess is y_p == E + D*x + A*x^4 + B*x^3 + C*x^2 Correct! y_p == - (1101*x)/625 - (307*x^2)/125 - (41*x^3)/25 - x^4/5 + 1166/3125 Correct! The solution is: y(x) == C1*exp(2*x)*cos(x) - C2*exp(2*x)*sin(x) That is correct! The general solution is y == - (1101*x)/625 - (307*x^2)/125 - (41*x^3)/25 - x^4/5 + C1*exp(2*x)*cos(x) - C2*exp(2*x)*sin(x) + 1166/3125 Solve y(x) - 2*diff(y(x), x) + diff(y(x), x, x) == 3*exp(-x) WorkOutGuess2 = A*exp(-x) Correct! The guess is y_p == A*exp(-x) Correct! y_p == (3*exp(-x))/4 ans = logical 1 Correct! The solution is: y(x) == C1*exp(x) + C2*x*exp(x) ans = logical 1 Solve 20*y(x) - 9*diff(y(x), x) + diff(y(x), x, x) == -2*cos(x) WorkOutGuess3 = B*cos(x) + A*sin(x) Correct! The guess is y_p == B*cos(x) + A*sin(x) Correct! y_p == -(442^(1/2)*cos(x + atan(9/19)))/221 ans = logical 1 Correct! The solution is: y(x) == C1*exp(4*x) + C2*exp(5*x) ans = logical 1 Solve 32*y(x) + 8*diff(y(x), x) + diff(y(x), x, x) == exp(-4*x)*(3*x^3*cos(4*x) + 4*x^4*sin(4*x)) WorkOutGuess4 = x*exp(-4*x)*(cos(4*x)*(A*x^4 + B*x^3 + C*x^2 + D*x + E) - sin(4*x)*(F*x^4 + G*x^3 + H*x^2 + I*x + J)) Technically correct! Conventionally, this guess is written with positive coefficients as y_p == x*exp(-4*x)*(cos(4*x)*(E + D*x + A*x^4 + B*x^3 + C*x^2) + sin(4*x)*(J + I*x + F*x^4 + G*x^3 + H*x^2)) Please try again. ans = logical 0 Correct! The solution is: y(x) == C1*cos(4*x)*exp(-4*x) - C2*sin(4*x)*exp(-4*x) ans = logical 1 Correct! The steady-state solution is x_p(t) == (F_0*t*sin(omega*t))/(2*(k*m)^(1/2)). Parameters: γ=0.22, ω₀=1.80, F₀=0.50, ω=3.60 -(3*exp(-(11*t)/100)*(2483*cos((13*191^(1/2)*t)/100) + 11*191^(1/2)*sin((13*191^(1/2)*t)/100)))/24830 Correct! The computed value for the spring coefficient for HCl is approximately 476 N/m. Bond wavenumber: 2886, External wavenumber: 2886, Response Amplitude: 1.8382e-15 [Terse] Diagnostic logged (2026-01-12 16:11:52): Figure saved to: --> /tmp/d6eeaa4f-d848-4515-a428-a5c6d6fe05f2/Figure_c4f16d1c-1f5a-4ee1-a4e9-a6d50ad4f52c.png [Terse] Diagnostic logged (2026-01-12 16:11:56): Figure saved to: --> /tmp/d6eeaa4f-d848-4515-a428-a5c6d6fe05f2/Figure_d080ade1-5aa2-450f-a944-00caa273b861.png . Done SolnSmokeTests __________