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″ + 7y′ + 10y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 7y′ + 10y = 0
r =
2.Easy
t (h)y5 h232

A quantity grows according to dy/dt = k·y with k > 0. It is 80 at t = 0 and 232 at t = 5 h. Find the rate constant k in h⁻¹.

y(0) = 80y(5) = 232
k = h⁻¹
3.Medium
t (min)T (°C)22Ta​T = ?

A can of soda at 5 °C is taken out of the fridge into a room at 26 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.09 min⁻¹, find its temperature T after 22 min.

T0 = 5 °CTa = 26 °Ck = 0.09 min⁻¹t = 22 min
T = °C
4.Medium
tyy(0) = 3y(1.5) = ?

Solve y″ + 7y′ + 12y = 0 with y(0) = 3 and y′(0) = 2. Find y(1.5).

y″ + 7y′ + 12y = 0y(0) = 3y′(0) = 2
y(1.5) =
5.Hard
t (min)m (kg)t = ?84 kg

A tank holds 500 L of water in which 20 kg of salt is dissolved. Brine with 0.2 kg of salt per litre flows in at 20 L/min, and the well-mixed solution flows out at the same rate. After how many minutes t does the tank contain 84 kg of salt?

V = 500 Lm0 = 20 kgc = 0.2 kg/Lr = 20 L/min
t = min
6.Hard
tyK = 1ymax​ = ?

A system obeys y″ + y′ + 9.25y = 9.25 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + y′ + 9.25y = 9.25y(0) = 0y′(0) = 0
ymax =

Answer key

  1. 1. r = −2
  2. 2. k = 0.2129 h⁻¹
  3. 3. T = 23.1 °C
  4. 4. y(1.5) = 0.1283
  5. 5. t = 40.24 min
  6. 6. ymax = 1.592

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