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

y″ + 7y′ + 6y = 0
r =
2.Easy
t (h)N20 hN = ?

A bacterial culture starts with 5000 cells and grows according to dN/dt = k·N with k = 0.17 h⁻¹. How many cells N are there after 20 h?

N0 = 5000k = 0.17 h⁻¹t = 20 h
N =
3.Medium
tyy(0) = 1y(1) = ?

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

y″ + 2y′ + y = 0y(0) = 1y′(0) = 0
y(1) =
4.Medium
tyy(0) = 2y(2) = ?

Solve y″ + 2y′ + y = 0 with y(0) = 2 and y′(0) = 2. Find y(2).

y″ + 2y′ + y = 0y(0) = 2y′(0) = 2
y(2) =
5.Hard
ty1y(1) = ?

Solve y′ + 3y = 7e−0.5t with y(0) = 5. Find y(1).

y′ + 3y = 7e−0.5ty(0) = 5
y(1) =
6.Hard
t (min)m (kg)t = ?41 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 5 L/min, and the well-mixed solution flows out at the same rate. After how many minutes t does the tank contain 41 kg of salt?

V = 500 Lm0 = 20 kgc = 0.1 kg/Lr = 5 L/min
t = min

Answer key

  1. 1. r = −1
  2. 2. N = 149821
  3. 3. y(1) = 0.7358
  4. 4. y(2) = 1.353
  5. 5. y(1) = 1.808
  6. 6. t = 120.4 min

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