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′ + 16y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 3.2y′ + 16y = 0
ζ =
2.Easy
t (days)m (mg)54 dm = ?

A radioactive tracer has a half-life of 20 days. A sample initially contains 120 mg. The decay follows dm/dt = −k·m. How much m remains after 54 days?

m0 = 120 mgt½ = 20 dt = 54 d
m = mg
3.Medium
t (min)T (°C)28Ta​24 °C

A can of soda at 3 °C is taken out of the fridge into a room at 29 °C. After 28 min its temperature is 24 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 3 °CTa = 29 °CT(28) = 24 °C
k = min⁻¹
4.Medium
tyy(0) = 3y(1.5) = ?

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

y″ + 2y′ + y = 0y(0) = 3y′(0) = 1
y(1.5) =
5.Hard
tyy(0) = 2y(0.75) = ?

Solve y″ + y′ + 9.25y = 0 with y(0) = 2 and y′(0) = 0 (released from rest). Find y(0.75); angles in radians.

y″ + y′ + 9.25y = 0y(0) = 2y′(0) = 0
y(0.75) =
6.Hard
tyK = 5ymax​ = ?

A system obeys y″ + 2y′ + 5y = 25 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + 2y′ + 5y = 25y(0) = 0y′(0) = 0
ymax =

Answer key

  1. 1. ζ = 0.4
  2. 2. m = 18.47 mg
  3. 3. k = 0.05888 min⁻¹
  4. 4. y(1.5) = 2.008
  5. 5. y(0.75) = −0.6852
  6. 6. ymax = 6.039

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