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
rr² + br + c?

For the equation y″ + 7y′ + 10y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 7y′ + 10y = 0
r =
3.Medium
tyy(0) = 3y(2) = ?

Solve y″ + 7y′ + 10y = 0 with y(0) = 3 and y′(0) = -4. Find y(2).

y″ + 7y′ + 10y = 0y(0) = 3y′(0) = -4
y(2) =
4.Medium
t (min)T (°C)17Ta​17 °C23 °C

A can of soda at 3 °C is taken out of the fridge into a room at 28 °C. After 17 min its temperature is 17 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 23 °C?

T0 = 3 °CTa = 28 °CT(17) = 17 °CT = 23 °C
t = min
5.Hard
ty2y(2) = ?

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

y′ + 3y = 3e−ty(0) = 0
y(2) =
6.Hard
tyy(0) = 10y(1) = ?

Solve y″ + 3y′ + 18.25y = 0 with y(0) = 10 and y′(0) = 0 (released from rest). Find y(1); angles in radians.

y″ + 3y′ + 18.25y = 0y(0) = 10y′(0) = 0
y(1) =

Answer key

  1. 1. r = −1
  2. 2. r = −2
  3. 3. y(2) = 0.06713
  4. 4. t = 33.33 min
  5. 5. y(2) = 0.1993
  6. 6. y(1) = −2.092

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