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″ + 3.6y′ + 9y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 3.6y′ + 9y = 0
ζ =
2.Easy
t (days)m (mg)79 dm = ?

A radioactive tracer has a half-life of 29 days. A sample initially contains 10 mg. The decay follows dm/dt = −k·m. How much m remains after 79 days?

m0 = 10 mgt½ = 29 dt = 79 d
m = mg
3.Medium
tyy(0) = 1y(1.5) = ?

Solve y″ + 3y′ + 2y = 0 with y(0) = 1 and y′(0) = -2. Find y(1.5).

y″ + 3y′ + 2y = 0y(0) = 1y′(0) = -2
y(1.5) =
4.Medium
t (min)T (°C)9Ta​14 °C

A can of soda at 6 °C is taken out of the fridge into a room at 24 °C. After 9 min its temperature is 14 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 6 °CTa = 24 °CT(9) = 14 °C
k = min⁻¹
5.Hard
ty1y(1) = ?

Solve y′ + y = 3e−0.5t with y(0) = 4. Find y(1).

y′ + y = 3e−0.5ty(0) = 4
y(1) =
6.Hard
tyy(0) = 10y(0.75) = ?

Solve y″ + 4y′ + 8y = 0 with y(0) = 10 and y′(0) = 0 (released from rest). Find y(0.75); angles in radians.

y″ + 4y′ + 8y = 0y(0) = 10y′(0) = 0
y(0.75) =

Answer key

  1. 1. ζ = 0.6
  2. 2. m = 1.513 mg
  3. 3. y(1.5) = 0.04979
  4. 4. k = 0.06531 min⁻¹
  5. 5. y(1) = 2.903
  6. 6. y(0.75) = 2.384

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