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″ + 3y′ + 25y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 3y′ + 25y = 0
ζ =
2.Easy
t (h)y4 h35

A quantity decays according to dy/dt = −k·y with k > 0. It is 50 at t = 0 and 35 at t = 4 h. Find the rate constant k in h⁻¹.

y(0) = 50y(4) = 35
k = h⁻¹
3.Medium
tyy(0) = 5y(0.5) = ?

Solve y″ + 8y′ + 15y = 0 with y(0) = 5 and y′(0) = 2. Find y(0.5).

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

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

y″ + 4y′ + 3y = 0y(0) = 4y′(0) = -4
y(0.5) =
5.Hard
t (min)m (kg)10 minm = ?

A tank holds 100 L of water in which 20 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 10 min?

V = 100 Lm0 = 20 kgc = 0.05 kg/Lr = 20 L/min
m = kg
6.Hard
tyK = 1y(1.5) = ?

A system obeys y″ + 4y′ + 29y = 29 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(1.5); angles in radians.

y″ + 4y′ + 29y = 29y(0) = 0y′(0) = 0
y(1.5) =

Answer key

  1. 1. ζ = 0.3
  2. 2. k = 0.08917 h⁻¹
  3. 3. y(0.5) = 2.315
  4. 4. y(0.5) = 2.426
  5. 5. m = 7.03 kg
  6. 6. y(1.5) = 0.9641

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