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)y9 h400

A quantity grows according to dy/dt = k·y with k > 0. It is 200 at t = 0 and 400 at t = 9 h. Find the rate constant k in h⁻¹.

y(0) = 200y(9) = 400
k = h⁻¹
2.Easy
t (days)m (mg)44 dm = ?

A radioactive tracer has a half-life of 25 days. A sample initially contains 10 mg. The decay follows dm/dt = −k·m. How much m remains after 44 days?

m0 = 10 mgt½ = 25 dt = 44 d
m = mg
3.Medium
tyy(0) = 2y(1.5) = ?

Solve y″ + 7y′ + 10y = 0 with y(0) = 2 and y′(0) = 4. Find y(1.5).

y″ + 7y′ + 10y = 0y(0) = 2y′(0) = 4
y(1.5) =
4.Medium
tyy(0) = 2y(1.5) = ?

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

y″ + 4y′ + 4y = 0y(0) = 2y′(0) = -2
y(1.5) =
5.Hard
tyK = 10y(0.5) = ?

A system obeys y″ + 2y′ + 5y = 50 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(0.5); angles in radians.

y″ + 2y′ + 5y = 50y(0) = 0y′(0) = 0
y(0.5) =
6.Hard
xy12(1, 0)y(2) = ?

Solve the initial value problem y′ + y/x = 4x2 for x > 0, with y(1) = 0. Find y(2).

y′ + y/x = 4x2y(1) = 0
y(2) =

Answer key

  1. 1. k = 0.07702 h⁻¹
  2. 2. m = 2.952 mg
  3. 3. y(1.5) = 0.2309
  4. 4. y(1.5) = 0.2489
  5. 5. y(0.5) = 4.171
  6. 6. y(2) = 7.5

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