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 (days)m (mg)61 dm = ?

A radioactive tracer has a half-life of 17 days. A sample initially contains 120 mg. The decay follows dm/dt = −k·m. How much m remains after 61 days?

m0 = 120 mgt½ = 17 dt = 61 d
m = mg
2.Easy
ty1

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

y″ + 2y′ + 4y = 0
ζ =
3.Medium
t (min)T (°C)21Ta​60 °C70 °C

A cup of coffee at 94 °C is left in a room at 21 °C. After 21 min its temperature is 60 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 70 °C?

T0 = 94 °CTa = 21 °CT(21) = 60 °CT = 70 °C
t = min
4.Medium
tyy(0) = 5y(1.5) = ?

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

y″ + 2y′ + y = 0y(0) = 5y′(0) = 4
y(1.5) =
5.Hard
tyy(0) = 10y(0.75) = ?

Solve y″ + 2y′ + 5y = 0 with y(0) = 10 and y′(0) = 0 (released from rest). Find y(0.75); angles in radians.

y″ + 2y′ + 5y = 0y(0) = 10y′(0) = 0
y(0.75) =
6.Hard
t (min)m (kg)t = ?64 kg

A tank holds 400 L of pure water. Brine with 0.2 kg of salt per litre flows in at 10 L/min, and the well-mixed solution flows out at the same rate. After how many minutes t does the tank contain 64 kg of salt?

V = 400 Lm0 = 0 kgc = 0.2 kg/Lr = 10 L/min
t = min

Answer key

  1. 1. m = 9.977 mg
  2. 2. ζ = 0.5
  3. 3. t = 13.35 min
  4. 4. y(1.5) = 4.128
  5. 5. y(0.75) = 2.69
  6. 6. t = 64.38 min

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