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″ + 3.2y′ + 64y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 3.2y′ + 64y = 0
ζ =
2.Easy
t (days)m (mg)22 dm = ?

A radioactive tracer has a half-life of 9 days. A sample initially contains 80 mg. The decay follows dm/dt = −k·m. How much m remains after 22 days?

m0 = 80 mgt½ = 9 dt = 22 d
m = mg
3.Medium
t (min)T (°C)7Ta​T = ?

A cup of coffee at 89 °C is left in a room at 20 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.04 min⁻¹, find its temperature T after 7 min.

T0 = 89 °CTa = 20 °Ck = 0.04 min⁻¹t = 7 min
T = °C
4.Medium
tyy(0) = 4y(1.5) = ?

Solve y″ + 5y′ + 6y = 0 with y(0) = 4 and y′(0) = 2. Find y(1.5).

y″ + 5y′ + 6y = 0y(0) = 4y′(0) = 2
y(1.5) =
5.Hard
tyK = 1y(0.25) = ?

A system obeys y″ + 4y′ + 8y = 8 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(0.25); angles in radians.

y″ + 4y′ + 8y = 8y(0) = 0y′(0) = 0
y(0.25) =
6.Hard
tyK = 2ymax​ = ?

A system obeys y″ + 4y′ + 20y = 40 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + 4y′ + 20y = 40y(0) = 0y′(0) = 0
ymax =

Answer key

  1. 1. ζ = 0.2
  2. 2. m = 14.7 mg
  3. 3. T = 72.15 °C
  4. 4. y(1.5) = 0.5859
  5. 5. y(0.25) = 0.1769
  6. 6. ymax = 2.416

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