Differential equations worksheet with answers

Free printable Differential equations worksheet: 6 problems from easy to exam level, with an answer key. Print it or save it as a PDF.

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1.Easy
rr² + br + c?

For the equation y″ + 9y′ + 18y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 9y′ + 18y = 0
r =
2.Easy
t (h)y12 h85

A quantity decays according to dy/dt = −k·y with k > 0. It is 100 at t = 0 and 85 at t = 12 h. Find the rate constant k in h⁻¹.

y(0) = 100y(12) = 85
k = h⁻¹
3.Medium
tyy(0) = 4y(2) = ?

Solve y″ + 7y′ + 10y = 0 with y(0) = 4 and y′(0) = 1. Find y(2).

y″ + 7y′ + 10y = 0y(0) = 4y′(0) = 1
y(2) =
4.Medium
tyy(0) = 3y(1.5) = ?

Solve y″ + 2y′ + y = 0 with y(0) = 3 and y′(0) = -1. Find y(1.5).

y″ + 2y′ + y = 0y(0) = 3y′(0) = -1
y(1.5) =
5.Hard
ty1.5y(1.5) = ?

Solve y′ + y = 2e−0.5t with y(0) = 3. Find y(1.5).

y′ + y = 2e−0.5ty(0) = 3
y(1.5) =
6.Hard
tyK = 10ymax​ = ?

A system obeys y″ + 4y′ + 13y = 130 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + 4y′ + 13y = 130y(0) = 0y′(0) = 0
ymax =

Answer key

  1. 1. r = −3
  2. 2. k = 0.01354 h⁻¹
  3. 3. y(2) = 0.1281
  4. 4. y(1.5) = 1.339
  5. 5. y(1.5) = 1.666
  6. 6. ymax = 11.23

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