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)2 dm = ?

A radioactive tracer has a half-life of 29 days. A sample initially contains 20 mg. The decay follows dm/dt = −k·m. How much m remains after 2 days?

m0 = 20 mgt½ = 29 dt = 2 d
m = mg
2.Easy
ty1

A mass–spring–damper is described by y″ + 2y′ + 100y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 2y′ + 100y = 0
ζ =
3.Medium
t (min)T (°C)28Ta​T = ?

A cup of coffee at 89 °C is left in a room at 18 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.05 min⁻¹, find its temperature T after 28 min.

T0 = 89 °CTa = 18 °Ck = 0.05 min⁻¹t = 28 min
T = °C
4.Medium
t (min)T (°C)22Ta​198 °C

A forged steel part at 360 °C is left to cool in a workshop at 24 °C. After 22 min its temperature is 198 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 360 °CTa = 24 °CT(22) = 198 °C
k = min⁻¹
5.Hard
tyK = 1y(1.5) = ?

A system obeys y″ + y′ + 25.25y = 25.25 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(1.5); angles in radians.

y″ + y′ + 25.25y = 25.25y(0) = 0y′(0) = 0
y(1.5) =
6.Hard
tyy(0) = 2y(1.5) = ?

Solve y″ + 4y′ + 20y = 0 with y(0) = 2 and y′(0) = 0 (released from rest). Find y(1.5); angles in radians.

y″ + 4y′ + 20y = 0y(0) = 2y′(0) = 0
y(1.5) =

Answer key

  1. 1. m = 19.07 mg
  2. 2. ζ = 0.1
  3. 3. T = 35.51 °C
  4. 4. k = 0.02991 min⁻¹
  5. 5. y(1.5) = 0.792
  6. 6. y(1.5) = 0.0817

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