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″ + 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 =
2.Easy
ty1

A mass–spring–damper is described by y″ + 18y′ + 36y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 18y′ + 36y = 0
ζ =
3.Medium
tyy(0) = 5y(2) = ?

Solve y″ + 4y′ + 3y = 0 with y(0) = 5 and y′(0) = -2. Find y(2).

y″ + 4y′ + 3y = 0y(0) = 5y′(0) = -2
y(2) =
4.Medium
tyy(0) = 5y(2) = ?

Solve y″ + 6y′ + 9y = 0 with y(0) = 5 and y′(0) = 4. Find y(2).

y″ + 6y′ + 9y = 0y(0) = 5y′(0) = 4
y(2) =
5.Hard
t (min)m (kg)5 minm = ?

A tank holds 400 L of water in which 5 kg of salt is dissolved. Brine with 0.05 kg of salt per litre flows in at 20 L/min, and the well-mixed solution flows out at the same rate. How much salt m is in the tank after 5 min?

V = 400 Lm0 = 5 kgc = 0.05 kg/Lr = 20 L/min
m = kg
6.Hard
ty0.5y(0.5) = ?

Solve y′ + 0.5y = 8e−2t with y(0) = 3. Find y(0.5).

y′ + 0.5y = 8e−2ty(0) = 3
y(0.5) =

Answer key

  1. 1. r = −1
  2. 2. ζ = 1.5
  3. 3. y(2) = 0.876
  4. 4. y(2) = 0.1066
  5. 5. m = 8.318 kg
  6. 6. y(0.5) = 4.528

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