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″ + 2.4y′ + 4y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 2.4y′ + 4y = 0
ζ =
2.Easy
t (h)N23 hN = ?

A bacterial culture starts with 200 cells and grows according to dN/dt = k·N with k = 0.38 h⁻¹. How many cells N are there after 23 h?

N0 = 200k = 0.38 h⁻¹t = 23 h
N =
3.Medium
t (min)T (°C)20Ta​267 °C134 °C

A forged steel part at 320 °C is left to cool in a workshop at 25 °C. After 20 min its temperature is 267 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 134 °C?

T0 = 320 °CTa = 25 °CT(20) = 267 °CT = 134 °C
t = min
4.Medium
tyy(0) = 2y(1.5) = ?

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

y″ + 2y′ + y = 0y(0) = 2y′(0) = -2
y(1.5) =
5.Hard
t (min)m (kg)75 minm = ?

A tank holds 500 L of pure water. Brine with 0.2 kg of salt per litre flows in at 4 L/min, and the well-mixed solution flows out at the same rate. How much salt m is in the tank after 75 min?

V = 500 Lm0 = 0 kgc = 0.2 kg/Lr = 4 L/min
m = kg
6.Hard
tyK = 1y(0.25) = ?

A system obeys y″ + 2y′ + 5y = 5 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(0.25); angles in radians.

y″ + 2y′ + 5y = 5y(0) = 0y′(0) = 0
y(0.25) =

Answer key

  1. 1. ζ = 0.6
  2. 2. N = 1.25 × 10⁶
  3. 3. t = 100.5 min
  4. 4. y(1.5) = 0.4463
  5. 5. m = 45.12 kg
  6. 6. y(0.25) = 0.1298

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