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′ + 4y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 5y′ + 4y = 0
r =
2.Easy
t (h)N3 hN = ?

A bacterial culture starts with 1500 cells and grows according to dN/dt = k·N with k = 0.04 h⁻¹. How many cells N are there after 3 h?

N0 = 1500k = 0.04 h⁻¹t = 3 h
N =
3.Medium
t (min)T (°C)17Ta​47 °C

A cup of coffee at 82 °C is left in a room at 21 °C. After 17 min its temperature is 47 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 82 °CTa = 21 °CT(17) = 47 °C
k = min⁻¹
4.Medium
tyy(0) = 1y(1) = ?

Solve y″ + 8y′ + 16y = 0 with y(0) = 1 and y′(0) = -2. Find y(1).

y″ + 8y′ + 16y = 0y(0) = 1y′(0) = -2
y(1) =
5.Hard
tyK = 1ymax​ = ?

A system obeys y″ + 4y′ + 8y = 8 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + 4y′ + 8y = 8y(0) = 0y′(0) = 0
ymax =
6.Hard
t (min)m (kg)t = ?32 kg

A tank holds 500 L of water in which 20 kg of salt is dissolved. Brine with 0.1 kg of salt per litre flows in at 10 L/min, and the well-mixed solution flows out at the same rate. After how many minutes t does the tank contain 32 kg of salt?

V = 500 Lm0 = 20 kgc = 0.1 kg/Lr = 10 L/min
t = min

Answer key

  1. 1. r = −1
  2. 2. N = 1691
  3. 3. k = 0.05016 min⁻¹
  4. 4. y(1) = 0.05495
  5. 5. ymax = 1.043
  6. 6. t = 25.54 min

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