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″ + 8.4y′ + 36y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 8.4y′ + 36y = 0
ζ =
2.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 =
3.Medium
tyy(0) = 3y(0.5) = ?

Solve y″ + 4y′ + 4y = 0 with y(0) = 3 and y′(0) = 2. Find y(0.5).

y″ + 4y′ + 4y = 0y(0) = 3y′(0) = 2
y(0.5) =
4.Medium
tyy(0) = 4y(1) = ?

Solve y″ + 6y′ + 8y = 0 with y(0) = 4 and y′(0) = 4. Find y(1).

y″ + 6y′ + 8y = 0y(0) = 4y′(0) = 4
y(1) =
5.Hard
xy12(1, −1)y(2) = ?

Solve the initial value problem y′ + 3y/x = x for x > 0, with y(1) = −1. Find y(2).

y′ + 3y/x = xy(1) = −1
y(2) =
6.Hard
t (min)m (kg)25 minm = ?

A tank holds 400 L of pure water. Brine with 0.25 kg of salt per litre flows in at 4 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 = 400 Lm0 = 0 kgc = 0.25 kg/Lr = 4 L/min
m = kg

Answer key

  1. 1. ζ = 0.7
  2. 2. r = −2
  3. 3. y(0.5) = 2.575
  4. 4. y(1) = 1.243
  5. 5. y(2) = 0.65
  6. 6. m = 22.12 kg

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