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
ty1

A mass–spring–damper is described by y″ + 8y′ + 25y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 8y′ + 25y = 0
ζ =
2.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 =
3.Medium
tyy(0) = 5y(0.5) = ?

Solve y″ + 8y′ + 15y = 0 with y(0) = 5 and y′(0) = 4. Find y(0.5).

y″ + 8y′ + 15y = 0y(0) = 5y′(0) = 4
y(0.5) =
4.Medium
tyy(0) = 5y(2) = ?

Solve y″ + 6y′ + 8y = 0 with y(0) = 5 and y′(0) = 4. Find y(2).

y″ + 6y′ + 8y = 0y(0) = 5y′(0) = 4
y(2) =
5.Hard
xy13(1, 1)y(3) = ?

Solve the initial value problem y′ + 3y/x = 5x for x > 0, with y(1) = 1. Find y(3).

y′ + 3y/x = 5xy(1) = 1
y(3) =
6.Hard
t (min)m (kg)t = ?3 kg

A tank holds 100 L of pure water. Brine with 0.05 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 3 kg of salt?

V = 100 Lm0 = 0 kgc = 0.05 kg/Lr = 10 L/min
t = min

Answer key

  1. 1. ζ = 0.8
  2. 2. r = −1
  3. 3. y(0.5) = 2.456
  4. 4. y(2) = 0.2174
  5. 5. y(3) = 9
  6. 6. t = 9.163 min

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