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
t (h)y4 h300

A quantity grows according to dy/dt = k·y with k > 0. It is 120 at t = 0 and 300 at t = 4 h. Find the rate constant k in h⁻¹.

y(0) = 120y(4) = 300
k = h⁻¹
2.Easy
rr² + br + c?

For the equation y″ + 7y′ + 12y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 7y′ + 12y = 0
r =
3.Medium
tyy(0) = 1y(1.5) = ?

Solve y″ + 8y′ + 16y = 0 with y(0) = 1 and y′(0) = 1. Find y(1.5).

y″ + 8y′ + 16y = 0y(0) = 1y′(0) = 1
y(1.5) =
4.Medium
tyy(0) = 1y(1.5) = ?

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

y″ + 2y′ + y = 0y(0) = 1y′(0) = 1
y(1.5) =
5.Hard
xy13(1, 0)y(3) = ?

Solve the initial value problem y′ + y/x = x for x > 0, with y(1) = 0. Find y(3).

y′ + y/x = xy(1) = 0
y(3) =
6.Hard
tyy(0) = 5y(1) = ?

Solve y″ + 4y′ + 29y = 0 with y(0) = 5 and y′(0) = 0 (released from rest). Find y(1); angles in radians.

y″ + 4y′ + 29y = 0y(0) = 5y′(0) = 0
y(1) =

Answer key

  1. 1. k = 0.2291 h⁻¹
  2. 2. r = −3
  3. 3. y(1.5) = 0.02107
  4. 4. y(1.5) = 0.8925
  5. 5. y(3) = 2.889
  6. 6. y(1) = −0.06761

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