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″ + 2.4y′ + 16y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 2.4y′ + 16y = 0
ζ =
2.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 =
3.Medium
tyy(0) = 1y(1) = ?

Solve y″ + 3y′ + 2y = 0 with y(0) = 1 and y′(0) = -4. Find y(1).

y″ + 3y′ + 2y = 0y(0) = 1y′(0) = -4
y(1) =
4.Medium
t (min)T (°C)7Ta​12 °C8 °C

A can of soda at 6 °C is taken out of the fridge into a room at 28 °C. After 7 min its temperature is 12 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 8 °C?

T0 = 6 °CTa = 28 °CT(7) = 12 °CT = 8 °C
t = min
5.Hard
t (min)m (kg)10 minm = ?

A tank holds 500 L of water in which 10 kg of salt is dissolved. Brine with 0.05 kg of salt per litre flows in at 2 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 = 500 Lm0 = 10 kgc = 0.05 kg/Lr = 2 L/min
m = kg
6.Hard
ty2y(2) = ?

Solve y′ + 3y = 10e−t with y(0) = 0. Find y(2).

y′ + 3y = 10e−ty(0) = 0
y(2) =

Answer key

  1. 1. ζ = 0.3
  2. 2. r = −1
  3. 3. y(1) = −0.3298
  4. 4. t = 2.095 min
  5. 5. m = 10.59 kg
  6. 6. y(2) = 0.6643

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