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″ + 10y′ + 100y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 10y′ + 100y = 0
ζ =
2.Easy
t (days)m (mg)9 dm = ?

A radioactive tracer has a half-life of 28 days. A sample initially contains 100 mg. The decay follows dm/dt = −k·m. How much m remains after 9 days?

m0 = 100 mgt½ = 28 dt = 9 d
m = mg
3.Medium
tyy(0) = 5y(1.5) = ?

Solve y″ + 7y′ + 10y = 0 with y(0) = 5 and y′(0) = -2. Find y(1.5).

y″ + 7y′ + 10y = 0y(0) = 5y′(0) = -2
y(1.5) =
4.Medium
t (min)T (°C)23Ta​158 °C100 °C

A forged steel part at 440 °C is left to cool in a workshop at 28 °C. After 23 min its temperature is 158 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 100 °C?

T0 = 440 °CTa = 28 °CT(23) = 158 °CT = 100 °C
t = min
5.Hard
t (min)m (kg)t = ?83 kg

A tank holds 500 L of water in which 20 kg of salt is dissolved. Brine with 0.25 kg of salt per litre flows in at 5 L/min, and the well-mixed solution flows out at the same rate. After how many minutes t does the tank contain 83 kg of salt?

V = 500 Lm0 = 20 kgc = 0.25 kg/Lr = 5 L/min
t = min
6.Hard
ty0.5y(0.5) = ?

Solve y′ + 3y = 5e−0.5t with y(0) = 2. Find y(0.5).

y′ + 3y = 5e−0.5ty(0) = 2
y(0.5) =

Answer key

  1. 1. ζ = 0.5
  2. 2. m = 80.03 mg
  3. 3. y(1.5) = 0.3802
  4. 4. t = 34.78 min
  5. 5. t = 91.63 min
  6. 6. y(0.5) = 1.558

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