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 (h)N15 hN = ?

A bacterial culture starts with 5000 cells and grows according to dN/dt = k·N with k = 0.34 h⁻¹. How many cells N are there after 15 h?

N0 = 5000k = 0.34 h⁻¹t = 15 h
N =
2.Easy
ty1

A mass–spring–damper is described by y″ + y′ + 25y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + y′ + 25y = 0
ζ =
3.Medium
tyy(0) = 2y(1.5) = ?

Solve y″ + 8y′ + 15y = 0 with y(0) = 2 and y′(0) = 2. Find y(1.5).

y″ + 8y′ + 15y = 0y(0) = 2y′(0) = 2
y(1.5) =
4.Medium
t (min)T (°C)40Ta​61 °C

A forged steel part at 320 °C is left to cool in a workshop at 20 °C. After 40 min its temperature is 61 °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(40) = 61 °C
k = min⁻¹
5.Hard
xy12(1, 4)y(2) = ?

Solve the initial value problem y′ + y/x = 3 for x > 0, with y(1) = 4. Find y(2).

y′ + y/x = 3y(1) = 4
y(2) =
6.Hard
t (min)m (kg)115 minm = ?

A tank holds 400 L of pure water. Brine with 0.25 kg of salt per litre flows in at 5 L/min, and the well-mixed solution flows out at the same rate. How much salt m is in the tank after 115 min?

V = 400 Lm0 = 0 kgc = 0.25 kg/Lr = 5 L/min
m = kg

Answer key

  1. 1. N = 820110
  2. 2. ζ = 0.1
  3. 3. y(1.5) = 0.06444
  4. 4. k = 0.04976 min⁻¹
  5. 5. y(2) = 4.25
  6. 6. m = 76.25 kg

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