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)y12 h180

A quantity grows according to dy/dt = k·y with k > 0. It is 100 at t = 0 and 180 at t = 12 h. Find the rate constant k in h⁻¹.

y(0) = 100y(12) = 180
k = h⁻¹
2.Easy
t (days)m (mg)84 dm = ?

A radioactive tracer has a half-life of 29 days. A sample initially contains 50 mg. The decay follows dm/dt = −k·m. How much m remains after 84 days?

m0 = 50 mgt½ = 29 dt = 84 d
m = mg
3.Medium
tyy(0) = 1y(2) = ?

Solve y″ + 5y′ + 4y = 0 with y(0) = 1 and y′(0) = 6. Find y(2).

y″ + 5y′ + 4y = 0y(0) = 1y′(0) = 6
y(2) =
4.Medium
tyy(0) = 3y(2) = ?

Solve y″ + 7y′ + 12y = 0 with y(0) = 3 and y′(0) = 2. Find y(2).

y″ + 7y′ + 12y = 0y(0) = 3y′(0) = 2
y(2) =
5.Hard
xy13(1, 0)y(3) = ?

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

y′ + 2y/x = 2xy(1) = 0
y(3) =
6.Hard
tyK = 5y(0.5) = ?

A system obeys y″ + 3y′ + 18.25y = 91.25 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(0.5); angles in radians.

y″ + 3y′ + 18.25y = 91.25y(0) = 0y′(0) = 0
y(0.5) =

Answer key

  1. 1. k = 0.04898 h⁻¹
  2. 2. m = 6.715 mg
  3. 3. y(2) = 0.4503
  4. 4. y(2) = 0.03101
  5. 5. y(3) = 4.444
  6. 6. y(0.5) = 5.178

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