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
t (h)y5 h700

A quantity grows according to dy/dt = k·y with k > 0. It is 200 at t = 0 and 700 at t = 5 h. Find the rate constant k in h⁻¹.

y(0) = 200y(5) = 700
k = h⁻¹
2.Easy
ty1

A mass–spring–damper is described by y″ + y′ + 25y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + y′ + 25y = 0
ζ =
3.Medium
tyy(0) = 1y(0.5) = ?

Solve y″ + 5y′ + 6y = 0 with y(0) = 1 and y′(0) = 0. Find y(0.5).

y″ + 5y′ + 6y = 0y(0) = 1y′(0) = 0
y(0.5) =
4.Medium
tyy(0) = 5y(1) = ?

Solve y″ + 6y′ + 9y = 0 with y(0) = 5 and y′(0) = -2. Find y(1).

y″ + 6y′ + 9y = 0y(0) = 5y′(0) = -2
y(1) =
5.Hard
xy13(1, −1)y(3) = ?

Solve the initial value problem y′ + 2y/x = 2x2 for x > 0, with y(1) = −1. Find y(3).

y′ + 2y/x = 2x2y(1) = −1
y(3) =
6.Hard
xy12(1, −3)y(2) = ?

Solve the initial value problem y′ + y/x = 1 for x > 0, with y(1) = −3. Find y(2).

y′ + y/x = 1y(1) = −3
y(2) =

Answer key

  1. 1. k = 0.2506 h⁻¹
  2. 2. ζ = 0.1
  3. 3. y(0.5) = 0.6574
  4. 4. y(1) = 0.8962
  5. 5. y(3) = 10.64
  6. 6. y(2) = −0.75

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