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″ + 8y′ + 15y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 8y′ + 15y = 0
r =
2.Easy
t (h)Nt = ?2500

A bacterial culture starts with 500 cells and grows according to dN/dt = k·N with k = 0.07 h⁻¹. After how many hours t are there 2500 cells?

N0 = 500k = 0.07 h⁻¹N = 2500
t = h
3.Medium
tyy(0) = 3y(0.5) = ?

Solve y″ + 4y′ + 4y = 0 with y(0) = 3 and y′(0) = 4. Find y(0.5).

y″ + 4y′ + 4y = 0y(0) = 3y′(0) = 4
y(0.5) =
4.Medium
t (min)T (°C)5Ta​7 °C

A can of soda at 4 °C is taken out of the fridge into a room at 23 °C. After 5 min its temperature is 7 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 4 °CTa = 23 °CT(5) = 7 °C
k = min⁻¹
5.Hard
t (min)m (kg)t = ?8 kg

A tank holds 200 L of pure water. Brine with 0.1 kg of salt per litre flows in at 2 L/min, and the well-mixed solution flows out at the same rate. After how many minutes t does the tank contain 8 kg of salt?

V = 200 Lm0 = 0 kgc = 0.1 kg/Lr = 2 L/min
t = min
6.Hard
tyK = 1ymax​ = ?

A system obeys y″ + 3y′ + 18.25y = 18.25 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + 3y′ + 18.25y = 18.25y(0) = 0y′(0) = 0
ymax =

Answer key

  1. 1. r = −3
  2. 2. t = 22.99 h
  3. 3. y(0.5) = 2.943
  4. 4. k = 0.03437 min⁻¹
  5. 5. t = 51.08 min
  6. 6. ymax = 1.308

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