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′ + 36y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 4.8y′ + 36y = 0
ζ =
2.Easy
t (days)m (mg)35 dm = ?

A radioactive tracer has a half-life of 14 days. A sample initially contains 20 mg. The decay follows dm/dt = −k·m. How much m remains after 35 days?

m0 = 20 mgt½ = 14 dt = 35 d
m = mg
3.Medium
t (min)T (°C)9Ta​52 °C79 °C

A cup of coffee at 91 °C is left in a room at 20 °C. After 9 min its temperature is 52 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 79 °C?

T0 = 91 °CTa = 20 °CT(9) = 52 °CT = 79 °C
t = min
4.Medium
t (min)T (°C)44Ta​59 °C

A forged steel part at 300 °C is left to cool in a workshop at 29 °C. After 44 min its temperature is 59 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 300 °CTa = 29 °CT(44) = 59 °C
k = min⁻¹
5.Hard
ty1.5y(1.5) = ?

Solve y′ + y = 3e−0.5t with y(0) = 4. Find y(1.5).

y′ + y = 3e−0.5ty(0) = 4
y(1.5) =
6.Hard
ty2y(2) = ?

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

y′ + 2y = 3e−0.5ty(0) = 0
y(2) =

Answer key

  1. 1. ζ = 0.4
  2. 2. m = 3.536 mg
  3. 3. t = 2.091 min
  4. 4. k = 0.05002 min⁻¹
  5. 5. y(1.5) = 2.388
  6. 6. y(2) = 0.6991

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