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)N8 hN = ?

A bacterial culture starts with 500 cells and grows according to dN/dt = k·N with k = 0.33 h⁻¹. How many cells N are there after 8 h?

N0 = 500k = 0.33 h⁻¹t = 8 h
N =
3.Medium
tyy(0) = 5y(2) = ?

Solve y″ + 8y′ + 16y = 0 with y(0) = 5 and y′(0) = 4. Find y(2).

y″ + 8y′ + 16y = 0y(0) = 5y′(0) = 4
y(2) =
4.Medium
tyy(0) = 5y(2) = ?

Solve y″ + 7y′ + 12y = 0 with y(0) = 5 and y′(0) = 6. Find y(2).

y″ + 7y′ + 12y = 0y(0) = 5y′(0) = 6
y(2) =
5.Hard
t (min)m (kg)25 minm = ?

A tank holds 500 L of water in which 5 kg of salt is dissolved. Brine with 0.2 kg of salt per litre flows in at 2 L/min, and the well-mixed solution flows out at the same rate. How much salt m is in the tank after 25 min?

V = 500 Lm0 = 5 kgc = 0.2 kg/Lr = 2 L/min
m = kg
6.Hard
tyK = 10ymax​ = ?

A system obeys y″ + 3y′ + 6.25y = 62.5 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + 3y′ + 6.25y = 62.5y(0) = 0y′(0) = 0
ymax =

Answer key

  1. 1. r = −2
  2. 2. N = 7007
  3. 3. y(2) = 0.01778
  4. 4. y(2) = 0.0574
  5. 5. m = 14.04 kg
  6. 6. ymax = 10.95

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