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″ + 4.8y′ + 9y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 4.8y′ + 9y = 0
ζ =
2.Easy
t (days)m (mg)28 dm = ?

A radioactive tracer has a half-life of 10 days. A sample initially contains 10 mg. The decay follows dm/dt = −k·m. How much m remains after 28 days?

m0 = 10 mgt½ = 10 dt = 28 d
m = mg
3.Medium
t (min)T (°C)15Ta​42 °C62 °C

A cup of coffee at 92 °C is left in a room at 21 °C. After 15 min its temperature is 42 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 62 °C?

T0 = 92 °CTa = 21 °CT(15) = 42 °CT = 62 °C
t = min
4.Medium
t (min)T (°C)25Ta​176 °C

A forged steel part at 440 °C is left to cool in a workshop at 22 °C. After 25 min its temperature is 176 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 440 °CTa = 22 °CT(25) = 176 °C
k = min⁻¹
5.Hard
t (min)m (kg)5 minm = ?

A tank holds 400 L of pure water. Brine with 0.05 kg of salt per litre flows in at 20 L/min, and the well-mixed solution flows out at the same rate. How much salt m is in the tank after 5 min?

V = 400 Lm0 = 0 kgc = 0.05 kg/Lr = 20 L/min
m = kg
6.Hard
xy13(1, −1)y(3) = ?

Solve the initial value problem y′ + y/x = 1 for x > 0, with y(1) = −1. Find y(3).

y′ + y/x = 1y(1) = −1
y(3) =

Answer key

  1. 1. ζ = 0.8
  2. 2. m = 1.436 mg
  3. 3. t = 6.762 min
  4. 4. k = 0.03994 min⁻¹
  5. 5. m = 4.424 kg
  6. 6. y(3) = 1

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