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)y2 h208

A quantity grows according to dy/dt = k·y with k > 0. It is 80 at t = 0 and 208 at t = 2 h. Find the rate constant k in h⁻¹.

y(0) = 80y(2) = 208
k = h⁻¹
2.Easy
t (h)y9 h675

A quantity grows according to dy/dt = k·y with k > 0. It is 250 at t = 0 and 675 at t = 9 h. Find the rate constant k in h⁻¹.

y(0) = 250y(9) = 675
k = h⁻¹
3.Medium
tyy(0) = 1y(1.5) = ?

Solve y″ + 3y′ + 2y = 0 with y(0) = 1 and y′(0) = 2. Find y(1.5).

y″ + 3y′ + 2y = 0y(0) = 1y′(0) = 2
y(1.5) =
4.Medium
tyy(0) = 3y(1.5) = ?

Solve y″ + 8y′ + 16y = 0 with y(0) = 3 and y′(0) = -2. Find y(1.5).

y″ + 8y′ + 16y = 0y(0) = 3y′(0) = -2
y(1.5) =
5.Hard
xy13(1, 5)y(3) = ?

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

y′ + 3y/x = 3x2y(1) = 5
y(3) =
6.Hard
ty1y(1) = ?

Solve y′ + 2y = 2e−3t with y(0) = 3. Find y(1).

y′ + 2y = 2e−3ty(0) = 3
y(1) =

Answer key

  1. 1. k = 0.4778 h⁻¹
  2. 2. k = 0.1104 h⁻¹
  3. 3. y(1.5) = 0.7432
  4. 4. y(1.5) = 0.04462
  5. 5. y(3) = 13.67
  6. 6. y(1) = 0.5771

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