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″ + 9.6y′ + 36y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 9.6y′ + 36y = 0
ζ =
2.Easy
t (days)m (mg)15 dm = ?

A radioactive tracer has a half-life of 9 days. A sample initially contains 80 mg. The decay follows dm/dt = −k·m. How much m remains after 15 days?

m0 = 80 mgt½ = 9 dt = 15 d
m = mg
3.Medium
t (min)T (°C)37Ta​67 °C

A forged steel part at 320 °C is left to cool in a workshop at 20 °C. After 37 min its temperature is 67 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 320 °CTa = 20 °CT(37) = 67 °C
k = min⁻¹
4.Medium
tyy(0) = 1y(1) = ?

Solve y″ + 4y′ + 4y = 0 with y(0) = 1 and y′(0) = 2. Find y(1).

y″ + 4y′ + 4y = 0y(0) = 1y′(0) = 2
y(1) =
5.Hard
tyy(0) = 1y(1) = ?

Solve y″ + y′ + 9.25y = 0 with y(0) = 1 and y′(0) = 0 (released from rest). Find y(1); angles in radians.

y″ + y′ + 9.25y = 0y(0) = 1y′(0) = 0
y(1) =
6.Hard
tyK = 10y(1) = ?

A system obeys y″ + 4y′ + 8y = 80 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(1); angles in radians.

y″ + 4y′ + 8y = 80y(0) = 0y′(0) = 0
y(1) =

Answer key

  1. 1. ζ = 0.8
  2. 2. m = 25.2 mg
  3. 3. k = 0.0501 min⁻¹
  4. 4. y(1) = 0.6767
  5. 5. y(1) = −0.5862
  6. 6. y(1) = 9.333

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