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″ + 5y′ + 4y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 5y′ + 4y = 0
r =
2.Easy
t (h)y2 h160

A quantity grows according to dy/dt = k·y with k > 0. It is 100 at t = 0 and 160 at t = 2 h. Find the rate constant k in h⁻¹.

y(0) = 100y(2) = 160
k = h⁻¹
3.Medium
tyy(0) = 2y(1.5) = ?

Solve y″ + 8y′ + 16y = 0 with y(0) = 2 and y′(0) = 0. Find y(1.5).

y″ + 8y′ + 16y = 0y(0) = 2y′(0) = 0
y(1.5) =
4.Medium
tyy(0) = 4y(0.5) = ?

Solve y″ + 7y′ + 10y = 0 with y(0) = 4 and y′(0) = 0. Find y(0.5).

y″ + 7y′ + 10y = 0y(0) = 4y′(0) = 0
y(0.5) =
5.Hard
tyy(0) = 2y(0.5) = ?

Solve y″ + 4y′ + 13y = 0 with y(0) = 2 and y′(0) = 0 (released from rest). Find y(0.5); angles in radians.

y″ + 4y′ + 13y = 0y(0) = 2y′(0) = 0
y(0.5) =
6.Hard
ty1.5y(1.5) = ?

Solve y′ + y = 2e−0.5t with y(0) = 4. Find y(1.5).

y′ + y = 2e−0.5ty(0) = 4
y(1.5) =

Answer key

  1. 1. r = −1
  2. 2. k = 0.235 h⁻¹
  3. 3. y(1.5) = 0.0347
  4. 4. y(0.5) = 2.234
  5. 5. y(0.5) = 0.5413
  6. 6. y(1.5) = 1.889

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