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″ + 6y′ + 5y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 6y′ + 5y = 0
r =
2.Easy
rr² + br + c?

For the equation y″ + 5y′ + 6y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 5y′ + 6y = 0
r =
3.Medium
tyy(0) = 2y(1) = ?

Solve y″ + 7y′ + 12y = 0 with y(0) = 2 and y′(0) = 1. Find y(1).

y″ + 7y′ + 12y = 0y(0) = 2y′(0) = 1
y(1) =
4.Medium
tyy(0) = 4y(1.5) = ?

Solve y″ + 7y′ + 12y = 0 with y(0) = 4 and y′(0) = 4. Find y(1.5).

y″ + 7y′ + 12y = 0y(0) = 4y′(0) = 4
y(1.5) =
5.Hard
tyy(0) = 5y(0.75) = ?

Solve y″ + 4y′ + 13y = 0 with y(0) = 5 and y′(0) = 0 (released from rest). Find y(0.75); angles in radians.

y″ + 4y′ + 13y = 0y(0) = 5y′(0) = 0
y(0.75) =
6.Hard
t (min)m (kg)t = ?28 kg

A tank holds 400 L of pure water. Brine with 0.1 kg of salt per litre flows in at 5 L/min, and the well-mixed solution flows out at the same rate. After how many minutes t does the tank contain 28 kg of salt?

V = 400 Lm0 = 0 kgc = 0.1 kg/Lr = 5 L/min
t = min

Answer key

  1. 1. r = −1
  2. 2. r = −2
  3. 3. y(1) = 0.3199
  4. 4. y(1.5) = 0.1825
  5. 5. y(0.75) = −0.1221
  6. 6. t = 96.32 min

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