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
rr² + br + c?

For the equation y″ + 5y′ + 6y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 5y′ + 6y = 0
r =
2.Easy
t (h)Nt = ?6000

A bacterial culture starts with 2000 cells and grows according to dN/dt = k·N with k = 0.23 h⁻¹. After how many hours t are there 6000 cells?

N0 = 2000k = 0.23 h⁻¹N = 6000
t = h
3.Medium
tyy(0) = 3y(1.5) = ?

Solve y″ + 8y′ + 16y = 0 with y(0) = 3 and y′(0) = -2. Find y(1.5).

y″ + 8y′ + 16y = 0y(0) = 3y′(0) = -2
y(1.5) =
4.Medium
t (min)T (°C)50Ta​T = ?

A forged steel part at 460 °C is left to cool in a workshop at 27 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.05 min⁻¹, find its temperature T after 50 min.

T0 = 460 °CTa = 27 °Ck = 0.05 min⁻¹t = 50 min
T = °C
5.Hard
tyK = 1ymax​ = ?

A system obeys y″ + y′ + 4.25y = 4.25 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + y′ + 4.25y = 4.25y(0) = 0y′(0) = 0
ymax =
6.Hard
t (min)m (kg)t = ?29 kg

A tank holds 1000 L of water in which 20 kg of salt is dissolved. Brine with 0.05 kg of salt per litre flows in at 4 L/min, and the well-mixed solution flows out at the same rate. After how many minutes t does the tank contain 29 kg of salt?

V = 1000 Lm0 = 20 kgc = 0.05 kg/Lr = 4 L/min
t = min

Answer key

  1. 1. r = −2
  2. 2. t = 4.777 h
  3. 3. y(1.5) = 0.04462
  4. 4. T = 62.54 °C
  5. 5. ymax = 1.456
  6. 6. t = 89.17 min

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