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″ + 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 =
2.Easy
t (h)N2 hN = ?

A bacterial culture starts with 2000 cells and grows according to dN/dt = k·N with k = 0.25 h⁻¹. How many cells N are there after 2 h?

N0 = 2000k = 0.25 h⁻¹t = 2 h
N =
3.Medium
tyy(0) = 5y(2) = ?

Solve y″ + 7y′ + 12y = 0 with y(0) = 5 and y′(0) = -4. Find y(2).

y″ + 7y′ + 12y = 0y(0) = 5y′(0) = -4
y(2) =
4.Medium
t (min)T (°C)25Ta​T = ?

A cup of coffee at 83 °C is left in a room at 18 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.05 min⁻¹, find its temperature T after 25 min.

T0 = 83 °CTa = 18 °Ck = 0.05 min⁻¹t = 25 min
T = °C
5.Hard
xy13(1, 1)y(3) = ?

Solve the initial value problem y′ + 2y/x = 5x2 for x > 0, with y(1) = 1. Find y(3).

y′ + 2y/x = 5x2y(1) = 1
y(3) =
6.Hard
tyy(0) = 1y(0.25) = ?

Solve y″ + 4y′ + 29y = 0 with y(0) = 1 and y′(0) = 0 (released from rest). Find y(0.25); angles in radians.

y″ + 4y′ + 29y = 0y(0) = 1y′(0) = 0
y(0.25) =

Answer key

  1. 1. r = −1
  2. 2. N = 3297
  3. 3. y(2) = 0.03597
  4. 4. T = 36.62 °C
  5. 5. y(3) = 27
  6. 6. y(0.25) = 0.4215

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