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
ty1

A mass–spring–damper is described by y″ + 4y′ + 100y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 4y′ + 100y = 0
ζ =
2.Easy
rr² + br + c?

For the equation y″ + 6y′ + 5y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 6y′ + 5y = 0
r =
3.Medium
tyy(0) = 3y(1.5) = ?

Solve y″ + 8y′ + 16y = 0 with y(0) = 3 and y′(0) = 4. Find y(1.5).

y″ + 8y′ + 16y = 0y(0) = 3y′(0) = 4
y(1.5) =
4.Medium
t (min)T (°C)58Ta​T = ?

A forged steel part at 360 °C is left to cool in a workshop at 29 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.01 min⁻¹, find its temperature T after 58 min.

T0 = 360 °CTa = 29 °Ck = 0.01 min⁻¹t = 58 min
T = °C
5.Hard
xy12(1, −3)y(2) = ?

Solve the initial value problem y′ + 3y/x = 2x2 for x > 0, with y(1) = −3. Find y(2).

y′ + 3y/x = 2x2y(1) = −3
y(2) =
6.Hard
tyK = 2ymax​ = ?

A system obeys y″ + 4y′ + 8y = 16 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + 4y′ + 8y = 16y(0) = 0y′(0) = 0
ymax =

Answer key

  1. 1. ζ = 0.2
  2. 2. r = −1
  3. 3. y(1.5) = 0.06693
  4. 4. T = 214.3 °C
  5. 5. y(2) = 2.25
  6. 6. ymax = 2.086

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