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Practice deliberately

Work through Differential Equations unit by unit, then test yourself under time.

Practice mode gives one hint and a worked explanation after every question. Test mode mixes all 7 units, holds the explanations until you submit, and then reports which units need another pass. The 8 questions printed further down this page are fixed, so you can read them and their worked answers without starting a set.

Differential Equations coverage and review map

Connect every question to the exact skill it rehearses. Work through the units in order, then return to any topic that still needs a hint or a second attempt.

This page follows the Differential Equations syllabus unit by unit. For shorter mixed retrieval outside the course sequence, use differential equations practice questions with answers.

01

Separable equations

Represent the constraints symbolically, choose an equivalence-preserving solution method, and verify the result in every original relationship.

Goal: recognise a separable equations problem from its wording, carry out the governing method, and check that the result is reasonable.

02

Linear first-order equations

Represent the constraints symbolically, choose an equivalence-preserving solution method, and verify the result in every original relationship.

Goal: recognise a linear first-order equations problem from its wording, carry out the governing method, and check that the result is reasonable.

03

Second-order equations

Represent the constraints symbolically, choose an equivalence-preserving solution method, and verify the result in every original relationship.

Goal: recognise a second-order equations problem from its wording, carry out the governing method, and check that the result is reasonable.

04

Laplace transforms

Define the central objects, connect at least two representations, solve direct and unfamiliar applications, and explain how the result can be checked.

Goal: recognise a laplace transforms problem from its wording, carry out the governing method, and check that the result is reasonable.

05

Systems

Represent the constraints symbolically, choose an equivalence-preserving solution method, and verify the result in every original relationship.

Goal: recognise a systems problem from its wording, carry out the governing method, and check that the result is reasonable.

06

Series solutions

Identify the generating pattern, distinguish term values from accumulated sums, and validate formulas with base cases and long-run behavior.

Goal: recognise a series solutions problem from its wording, carry out the governing method, and check that the result is reasonable.

07

Qualitative analysis

Define the central objects, connect at least two representations, solve direct and unfamiliar applications, and explain how the result can be checked.

Goal: recognise a qualitative analysis problem from its wording, carry out the governing method, and check that the result is reasonable.

Differential Equations questions with worked answers

These 8 questions are printed in full on this page and are drawn across all 7 course units. Nothing here is generated on the fly. Cover the options, solve each one on paper, and only then open the worked answer to compare your method with the one shown.

Question 1: Solve dy/dx = 3x^2 y with y(0) = 2.

Differential Equations · Separable equations

  1. y = 2 e^(x^3)
  2. y = 2 e^(3x)
  3. y = 2 + x^3
  4. y = e^(x^3) + 1
Show the worked answer
  1. Separate the variables: dy / y = 3x^2 dx.
  2. Integrate both sides: ln|y| = x^3 + C.
  3. Exponentiate: y = A e^(x^3), where A is a constant.
  4. At x = 0, e^0 = 1 so A = 2, giving y = 2 e^(x^3).

Answer: y = 2 e^(x^3)

Question 2: Solve y' + 2y = 6 with y(0) = 1, then state the long-run value of y.

Differential Equations · Linear first-order equations

  1. y = 3 - 2 e^(-2x), approaching 3
  2. y = 3 + 2 e^(-2x), approaching 3
  3. y = 6 - 5 e^(-2x), approaching 6
  4. y = 1 + 5x, growing without bound
Show the worked answer
  1. The equilibrium solution comes from setting y' = 0: 2y = 6, so y = 3.
  2. The general solution is 3 plus a multiple of e^(-2x).
  3. At x = 0: 3 + A = 1, so A = -2.
  4. y = 3 - 2 e^(-2x). The exponential dies out, so y approaches 3.

Answer: y = 3 - 2 e^(-2x), approaching 3

Question 3: What is the general solution of y'' - 5y' + 6y = 0?

Differential Equations · Second-order equations

  1. y = C1 e^(2x) + C2 e^(3x)
  2. y = C1 e^(-2x) + C2 e^(-3x)
  3. y = C1 e^(5x) + C2 e^(6x)
  4. y = (C1 + C2 x) e^(3x)
Show the worked answer
  1. Write the characteristic equation: r^2 - 5r + 6 = 0.
  2. Factor it: (r - 2)(r - 3) = 0.
  3. The roots are r = 2 and r = 3, and they are different real numbers.
  4. So the general solution is y = C1 e^(2x) + C2 e^(3x).

Answer: y = C1 e^(2x) + C2 e^(3x)

Question 4: Solve y'' + 4y = 0 with y(0) = 3 and y'(0) = 8, then give the amplitude.

Differential Equations · Second-order equations

  1. y = 3 cos(2x) + 4 sin(2x), amplitude 5
  2. y = 3 cos(2x) + 8 sin(2x), amplitude 11
  3. y = 3 cos(4x) + 2 sin(4x), amplitude 5
  4. y = 3 e^(2x) + 4 e^(-2x), amplitude 7
Show the worked answer
  1. The characteristic roots are plus and minus 2i, so y = A cos(2x) + B sin(2x).
  2. y(0) = A = 3.
  3. y' = -2A sin(2x) + 2B cos(2x), so y'(0) = 2B = 8 and B = 4.
  4. The amplitude is the square root of 3^2 + 4^2 = 25, which is 5.

Answer: y = 3 cos(2x) + 4 sin(2x), amplitude 5

Question 5: What is the Laplace transform of t times e^(3t)?

Differential Equations · Laplace transforms

  1. 1 / (s - 3)^2
  2. 1 / (s - 3)
  3. 1 / (s + 3)^2
  4. s / (s - 3)^2
Show the worked answer
  1. The transform of t alone is 1 / s^2.
  2. Multiplying by e^(at) shifts s to s - a in the transform.
  3. Here a = 3.
  4. So the transform is 1 / (s - 3)^2, valid for s greater than 3.

Answer: 1 / (s - 3)^2

Question 6: For the system X' = AX with A = [[1, 2], [3, 2]], what are the eigenvalues and what do they say about the origin?

Differential Equations · Systems

  1. 4 and -1, a saddle point
  2. 4 and 1, an unstable node
  3. -4 and -1, a stable node
  4. 3 and 2, a saddle point
Show the worked answer
  1. The trace is 1 + 2 = 3 and the determinant is 1(2) - 2(3) = -4.
  2. The characteristic equation is lambda^2 - 3 lambda - 4 = 0.
  3. It factors as (lambda - 4)(lambda + 1), so the eigenvalues are 4 and -1.
  4. Real eigenvalues with opposite signs make the origin a saddle point.

Answer: 4 and -1, a saddle point

Question 7: A power series solution of y'' = xy has a(0) = 1 and a(1) = 0. What are a(2) and a(3)?

Differential Equations · Series solutions

  1. a(2) = 0 and a(3) = 1/6
  2. a(2) = 1/2 and a(3) = 0
  3. a(2) = 0 and a(3) = 1/3
  4. a(2) = 1 and a(3) = 1/6
Show the worked answer
  1. Write y as a sum of a(n) x^n. Then y'' has constant term 2 a(2).
  2. On the right, xy has no constant term, so 2 a(2) = 0 and a(2) = 0.
  3. Matching the x terms: 6 a(3) = a(0) = 1.
  4. So a(3) = 1/6.

Answer: a(2) = 0 and a(3) = 1/6

Question 8: For dy/dt = y(4 - y), classify the equilibria and say what happens to a solution starting at y = 1.

Differential Equations · Qualitative analysis

  1. 0 is unstable and 4 is stable, so y rises toward 4
  2. 0 is stable and 4 is unstable, so y falls toward 0
  3. both are stable, so y stays at 1
  4. there are no equilibria, so y grows without bound
Show the worked answer
  1. Set the right side to zero: y(4 - y) = 0 gives equilibria at y = 0 and y = 4.
  2. For y between 0 and 4, both factors are positive, so dy/dt is positive and y increases.
  3. For y above 4, the factor (4 - y) is negative, so y decreases back toward 4.
  4. That makes 0 unstable and 4 stable, so a solution starting at y = 1 rises toward 4.

Answer: 0 is unstable and 4 is stable, so y rises toward 4

Which unit each printed question belongs to

Use this map after marking your work. If two misses share a unit, review that unit before starting a generated set.

The 8 printed Differential Equations questions, the unit each one tests, and its answer.
QuestionUnitWhat it asksAnswer
1Separable equationsSolve dy/dx = 3x^2 y with y(0) = 2.y = 2 e^(x^3)
2Linear first-order equationsSolve y' + 2y = 6 with y(0) = 1, then state the long-run value of y.y = 3 - 2 e^(-2x), approaching 3
3Second-order equationsWhat is the general solution of y'' - 5y' + 6y = 0?y = C1 e^(2x) + C2 e^(3x)
4Second-order equationsSolve y'' + 4y = 0 with y(0) = 3 and y'(0) = 8, then give the amplitude.y = 3 cos(2x) + 4 sin(2x), amplitude 5
5Laplace transformsWhat is the Laplace transform of t times e^(3t)?1 / (s - 3)^2
6SystemsFor the system X' = AX with A = [[1, 2], [3, 2]], what are the eigenvalues and what do they say about the origin?4 and -1, a saddle point
7Series solutionsA power series solution of y'' = xy has a(0) = 1 and a(1) = 0. What are a(2) and a(3)?a(2) = 0 and a(3) = 1/6
8Qualitative analysisFor dy/dt = y(4 - y), classify the equilibria and say what happens to a solution starting at y = 1.0 is unstable and 4 is stable, so y rises toward 4

How the generated Differential Equations sets are created

The 8 questions above are fixed and checked. The generator at the top of the page is different: it writes fresh questions with AI from the course and unit information shown here, then the application checks each one for a complete prompt, four choices, one keyed answer, and an explanation. Generated questions are original practice—not official or released exam questions—and AI can still make mathematical mistakes. Verify a disputed answer with the stated method, your course materials, or the MathGPT solver, and follow the site's academic-integrity guidance.

Questions about this Differential Equations practice page

What does this page cover?

It covers all 7 Differential Equations units listed above. Choose one unit for focused work, or mixed review to test method selection, and switch to test mode when you want all units mixed under time.

When can I see correct answers and explanations?

The 8 printed questions on this page keep their worked answers behind a toggle, so you can check any one of them straight away. In the generator above, practice mode explains each question as soon as you answer it, while test mode holds every explanation until you submit.

What should I do with a missed question?

Classify the miss as a definition, setup, calculation, interpretation, or timing error. Re-solve it from a blank page, then use the MathGPT tutor for a hint or method check.