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″ + 4.8y′ + 16y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 4.8y′ + 16y = 0
ζ =
2.Easy
rr² + br + c?

For the equation y″ + 4y′ + 3y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

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

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

y″ + 7y′ + 12y = 0y(0) = 2y′(0) = 6
y(2) =
4.Medium
tyy(0) = 5y(1.5) = ?

Solve y″ + 5y′ + 6y = 0 with y(0) = 5 and y′(0) = 4. Find y(1.5).

y″ + 5y′ + 6y = 0y(0) = 5y′(0) = 4
y(1.5) =
5.Hard
tyK = 10y(0.5) = ?

A system obeys y″ + y′ + 25.25y = 252.5 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(0.5); angles in radians.

y″ + y′ + 25.25y = 252.5y(0) = 0y′(0) = 0
y(0.5) =
6.Hard
t (min)m (kg)35 minm = ?

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 10 L/min, and the well-mixed solution flows out at the same rate. How much salt m is in the tank after 35 min?

V = 500 Lm0 = 20 kgc = 0.1 kg/Lr = 10 L/min
m = kg

Answer key

  1. 1. ζ = 0.6
  2. 2. r = −1
  3. 3. y(2) = 0.03068
  4. 4. y(1.5) = 0.7904
  5. 5. y(0.5) = 15.77
  6. 6. m = 35.1 kg

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