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 (days)m (mg)13 dm = ?

A radioactive tracer has a half-life of 4 days. A sample initially contains 10 mg. The decay follows dm/dt = −k·m. How much m remains after 13 days?

m0 = 10 mgt½ = 4 dt = 13 d
m = mg
2.Easy
rr² + br + c?

For the equation y″ + 6y′ + 8y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 6y′ + 8y = 0
r =
3.Medium
tyy(0) = 3y(0.5) = ?

Solve y″ + 4y′ + 4y = 0 with y(0) = 3 and y′(0) = -2. Find y(0.5).

y″ + 4y′ + 4y = 0y(0) = 3y′(0) = -2
y(0.5) =
4.Medium
tyy(0) = 2y(1) = ?

Solve y″ + 6y′ + 8y = 0 with y(0) = 2 and y′(0) = -2. Find y(1).

y″ + 6y′ + 8y = 0y(0) = 2y′(0) = -2
y(1) =
5.Hard
xy12(1, −2)y(2) = ?

Solve the initial value problem y′ + 3y/x = 4 for x > 0, with y(1) = −2. Find y(2).

y′ + 3y/x = 4y(1) = −2
y(2) =
6.Hard
tyy(0) = 1y(0.25) = ?

Solve y″ + 4y′ + 8y = 0 with y(0) = 1 and y′(0) = 0 (released from rest). Find y(0.25); angles in radians.

y″ + 4y′ + 8y = 0y(0) = 1y′(0) = 0
y(0.25) =

Answer key

  1. 1. m = 1.051 mg
  2. 2. r = −2
  3. 3. y(0.5) = 1.839
  4. 4. y(1) = 0.3877
  5. 5. y(2) = 1.625
  6. 6. y(0.25) = 0.8231

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