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″ + 3y′ + 2y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 3y′ + 2y = 0
r =
2.Easy
ty1

A mass–spring–damper is described by y″ + 7.2y′ + 36y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 7.2y′ + 36y = 0
ζ =
3.Medium
t (min)T (°C)33Ta​T = ?

A forged steel part at 400 °C is left to cool in a workshop at 28 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.04 min⁻¹, find its temperature T after 33 min.

T0 = 400 °CTa = 28 °Ck = 0.04 min⁻¹t = 33 min
T = °C
4.Medium
tyy(0) = 5y(2) = ?

Solve y″ + 4y′ + 4y = 0 with y(0) = 5 and y′(0) = 2. Find y(2).

y″ + 4y′ + 4y = 0y(0) = 5y′(0) = 2
y(2) =
5.Hard
xy12(1, −2)y(2) = ?

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

y′ + y/x = 2xy(1) = −2
y(2) =
6.Hard
xy12(1, 2)y(2) = ?

Solve the initial value problem y′ + y/x = 4x2 for x > 0, with y(1) = 2. Find y(2).

y′ + y/x = 4x2y(1) = 2
y(2) =

Answer key

  1. 1. r = −1
  2. 2. ζ = 0.6
  3. 3. T = 127.4 °C
  4. 4. y(2) = 0.5312
  5. 5. y(2) = 1.333
  6. 6. y(2) = 8.5

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