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)N20 hN = ?

A bacterial culture starts with 5000 cells and grows according to dN/dt = k·N with k = 0.18 h⁻¹. How many cells N are there after 20 h?

N0 = 5000k = 0.18 h⁻¹t = 20 h
N =
2.Easy
t (h)N20 hN = ?

A bacterial culture starts with 5000 cells and grows according to dN/dt = k·N with k = 0.17 h⁻¹. How many cells N are there after 20 h?

N0 = 5000k = 0.17 h⁻¹t = 20 h
N =
3.Medium
tyy(0) = 5y(1) = ?

Solve y″ + 8y′ + 15y = 0 with y(0) = 5 and y′(0) = 2. Find y(1).

y″ + 8y′ + 15y = 0y(0) = 5y′(0) = 2
y(1) =
4.Medium
tyy(0) = 2y(0.5) = ?

Solve y″ + 2y′ + y = 0 with y(0) = 2 and y′(0) = -1. Find y(0.5).

y″ + 2y′ + y = 0y(0) = 2y′(0) = -1
y(0.5) =
5.Hard
xy13(1, 2)y(3) = ?

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

y′ + y/x = 2xy(1) = 2
y(3) =
6.Hard
ty2y(2) = ?

Solve y′ + 2y = 7e−t with y(0) = 2. Find y(2).

y′ + 2y = 7e−ty(0) = 2
y(2) =

Answer key

  1. 1. N = 182991
  2. 2. N = 149821
  3. 3. y(1) = 0.6149
  4. 4. y(0.5) = 1.516
  5. 5. y(3) = 6.444
  6. 6. y(2) = 0.8558

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