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)y6 h1950

A quantity grows according to dy/dt = k·y with k > 0. It is 500 at t = 0 and 1950 at t = 6 h. Find the rate constant k in h⁻¹.

y(0) = 500y(6) = 1950
k = h⁻¹
2.Easy
t (days)m (mg)16 dm = ?

A radioactive tracer has a half-life of 7 days. A sample initially contains 50 mg. The decay follows dm/dt = −k·m. How much m remains after 16 days?

m0 = 50 mgt½ = 7 dt = 16 d
m = mg
3.Medium
tyy(0) = 4y(1.5) = ?

Solve y″ + 6y′ + 5y = 0 with y(0) = 4 and y′(0) = -4. Find y(1.5).

y″ + 6y′ + 5y = 0y(0) = 4y′(0) = -4
y(1.5) =
4.Medium
tyy(0) = 1y(2) = ?

Solve y″ + 6y′ + 8y = 0 with y(0) = 1 and y′(0) = 0. Find y(2).

y″ + 6y′ + 8y = 0y(0) = 1y′(0) = 0
y(2) =
5.Hard
xy13(1, 1)y(3) = ?

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

y′ + 2y/x = 3y(1) = 1
y(3) =
6.Hard
tyy(0) = 10y(0.5) = ?

Solve y″ + y′ + 4.25y = 0 with y(0) = 10 and y′(0) = 0 (released from rest). Find y(0.5); angles in radians.

y″ + y′ + 4.25y = 0y(0) = 10y′(0) = 0
y(0.5) =

Answer key

  1. 1. k = 0.2268 h⁻¹
  2. 2. m = 10.25 mg
  3. 3. y(1.5) = 0.8925
  4. 4. y(2) = 0.0363
  5. 5. y(3) = 3
  6. 6. y(0.5) = 5.846

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