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 (h)y12 h216

A quantity grows according to dy/dt = k·y with k > 0. It is 120 at t = 0 and 216 at t = 12 h. Find the rate constant k in h⁻¹.

y(0) = 120y(12) = 216
k = h⁻¹
2.Easy
ty1

A mass–spring–damper is described by y″ + 10y′ + 100y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 10y′ + 100y = 0
ζ =
3.Medium
t (min)T (°C)59Ta​199 °C

A forged steel part at 340 °C is left to cool in a workshop at 24 °C. After 59 min its temperature is 199 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 340 °CTa = 24 °CT(59) = 199 °C
k = min⁻¹
4.Medium
tyy(0) = 4y(1) = ?

Solve y″ + 7y′ + 12y = 0 with y(0) = 4 and y′(0) = 0. Find y(1).

y″ + 7y′ + 12y = 0y(0) = 4y′(0) = 0
y(1) =
5.Hard
ty1y(1) = ?

Solve y′ + y = 5e−0.5t with y(0) = 0. Find y(1).

y′ + y = 5e−0.5ty(0) = 0
y(1) =
6.Hard
tyy(0) = 5y(0.75) = ?

Solve y″ + 2y′ + 26y = 0 with y(0) = 5 and y′(0) = 0 (released from rest). Find y(0.75); angles in radians.

y″ + 2y′ + 26y = 0y(0) = 5y′(0) = 0
y(0.75) =

Answer key

  1. 1. k = 0.04898 h⁻¹
  2. 2. ζ = 0.5
  3. 3. k = 0.01002 min⁻¹
  4. 4. y(1) = 0.5768
  5. 5. y(1) = 2.387
  6. 6. y(0.75) = −2.208

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