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″ + 7.2y′ + 36y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 7.2y′ + 36y = 0
ζ =
2.Easy
rr² + br + c?

For the equation y″ + 7y′ + 12y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 7y′ + 12y = 0
r =
3.Medium
tyy(0) = 1y(2) = ?

Solve y″ + 4y′ + 4y = 0 with y(0) = 1 and y′(0) = -2. Find y(2).

y″ + 4y′ + 4y = 0y(0) = 1y′(0) = -2
y(2) =
4.Medium
tyy(0) = 5y(1.5) = ?

Solve y″ + 2y′ + y = 0 with y(0) = 5 and y′(0) = 4. Find y(1.5).

y″ + 2y′ + y = 0y(0) = 5y′(0) = 4
y(1.5) =
5.Hard
tyy(0) = 5y(1.5) = ?

Solve y″ + 2y′ + 26y = 0 with y(0) = 5 and y′(0) = 0 (released from rest). Find y(1.5); angles in radians.

y″ + 2y′ + 26y = 0y(0) = 5y′(0) = 0
y(1.5) =
6.Hard
t (min)m (kg)75 minm = ?

A tank holds 400 L of water in which 10 kg of salt is dissolved. Brine with 0.25 kg of salt per litre flows in at 10 L/min, and the well-mixed solution flows out at the same rate. How much salt m is in the tank after 75 min?

V = 400 Lm0 = 10 kgc = 0.25 kg/Lr = 10 L/min
m = kg

Answer key

  1. 1. ζ = 0.6
  2. 2. r = −3
  3. 3. y(2) = 0.01832
  4. 4. y(1.5) = 4.128
  5. 5. y(1.5) = 0.596
  6. 6. m = 86.2 kg

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