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″ + 7y′ + 6y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 7y′ + 6y = 0
r =
2.Easy
t (h)N23 hN = ?

A bacterial culture starts with 500 cells and grows according to dN/dt = k·N with k = 0.04 h⁻¹. How many cells N are there after 23 h?

N0 = 500k = 0.04 h⁻¹t = 23 h
N =
3.Medium
tyy(0) = 4y(0.5) = ?

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

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

Solve y″ + 8y′ + 15y = 0 with y(0) = 2 and y′(0) = 6. Find y(2).

y″ + 8y′ + 15y = 0y(0) = 2y′(0) = 6
y(2) =
5.Hard
tyy(0) = 1y(1.5) = ?

Solve y″ + 2y′ + 10y = 0 with y(0) = 1 and y′(0) = 0 (released from rest). Find y(1.5); angles in radians.

y″ + 2y′ + 10y = 0y(0) = 1y′(0) = 0
y(1.5) =
6.Hard
xy13(1, −1)y(3) = ?

Solve the initial value problem y′ + 3y/x = 4x2 for x > 0, with y(1) = −1. Find y(3).

y′ + 3y/x = 4x2y(1) = −1
y(3) =

Answer key

  1. 1. r = −1
  2. 2. N = 1255
  3. 3. y(0.5) = 1.853
  4. 4. y(2) = 0.01956
  5. 5. y(1.5) = −0.1197
  6. 6. y(3) = 17.94

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