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)N21 hN = ?

A bacterial culture starts with 5000 cells and grows according to dN/dt = k·N with k = 0.02 h⁻¹. How many cells N are there after 21 h?

N0 = 5000k = 0.02 h⁻¹t = 21 h
N =
3.Medium
t (min)T (°C)16Ta​21 °C23 °C

A can of soda at 5 °C is taken out of the fridge into a room at 25 °C. After 16 min its temperature is 21 °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 = 5 °CTa = 25 °CT(16) = 21 °CT = 23 °C
t = min
4.Medium
tyy(0) = 2y(1) = ?

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

y″ + 4y′ + 4y = 0y(0) = 2y′(0) = -1
y(1) =
5.Hard
ty1.5y(1.5) = ?

Solve y′ + y = 8e−0.5t with y(0) = 1. Find y(1.5).

y′ + y = 8e−0.5ty(0) = 1
y(1.5) =
6.Hard
xy12(1, 3)y(2) = ?

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

y′ + 2y/x = 2x2y(1) = 3
y(2) =

Answer key

  1. 1. r = −1
  2. 2. N = 7610
  3. 3. t = 22.89 min
  4. 4. y(1) = 0.6767
  5. 5. y(1.5) = 4.211
  6. 6. y(2) = 3.85

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