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)y4 h520

A quantity grows according to dy/dt = k·y with k > 0. It is 400 at t = 0 and 520 at t = 4 h. Find the rate constant k in h⁻¹.

y(0) = 400y(4) = 520
k = h⁻¹
2.Easy
t (h)y4 h1040

A quantity grows according to dy/dt = k·y with k > 0. It is 400 at t = 0 and 1040 at t = 4 h. Find the rate constant k in h⁻¹.

y(0) = 400y(4) = 1040
k = h⁻¹
3.Medium
tyy(0) = 5y(1) = ?

Solve y″ + 8y′ + 16y = 0 with y(0) = 5 and y′(0) = -2. Find y(1).

y″ + 8y′ + 16y = 0y(0) = 5y′(0) = -2
y(1) =
4.Medium
tyy(0) = 3y(1.5) = ?

Solve y″ + 3y′ + 2y = 0 with y(0) = 3 and y′(0) = -4. Find y(1.5).

y″ + 3y′ + 2y = 0y(0) = 3y′(0) = -4
y(1.5) =
5.Hard
xy13(1, 3)y(3) = ?

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

y′ + 3y/x = 3y(1) = 3
y(3) =
6.Hard
tyK = 5y(0.75) = ?

A system obeys y″ + 2y′ + 17y = 85 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(0.75); angles in radians.

y″ + 2y′ + 17y = 85y(0) = 0y′(0) = 0
y(0.75) =

Answer key

  1. 1. k = 0.06559 h⁻¹
  2. 2. k = 0.2389 h⁻¹
  3. 3. y(1) = 0.4213
  4. 4. y(1.5) = 0.496
  5. 5. y(3) = 2.333
  6. 6. y(0.75) = 7.255

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