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 (days)m (mg)61 dm = ?

A radioactive tracer has a half-life of 21 days. A sample initially contains 200 mg. The decay follows dm/dt = −k·m. How much m remains after 61 days?

m0 = 200 mgt½ = 21 dt = 61 d
m = mg
2.Easy
t (days)m (mg)32 dm = ?

A radioactive tracer has a half-life of 20 days. A sample initially contains 50 mg. The decay follows dm/dt = −k·m. How much m remains after 32 days?

m0 = 50 mgt½ = 20 dt = 32 d
m = mg
3.Medium
tyy(0) = 2y(0.5) = ?

Solve y″ + 8y′ + 15y = 0 with y(0) = 2 and y′(0) = 4. Find y(0.5).

y″ + 8y′ + 15y = 0y(0) = 2y′(0) = 4
y(0.5) =
4.Medium
tyy(0) = 5y(2) = ?

Solve y″ + 7y′ + 12y = 0 with y(0) = 5 and y′(0) = -2. Find y(2).

y″ + 7y′ + 12y = 0y(0) = 5y′(0) = -2
y(2) =
5.Hard
tyy(0) = 1y(0.75) = ?

Solve y″ + 3y′ + 11.25y = 0 with y(0) = 1 and y′(0) = 0 (released from rest). Find y(0.75); angles in radians.

y″ + 3y′ + 11.25y = 0y(0) = 1y′(0) = 0
y(0.75) =
6.Hard
tyy(0) = 1y(1) = ?

Solve y″ + y′ + 4.25y = 0 with y(0) = 1 and y′(0) = 0 (released from rest). Find y(1); angles in radians.

y″ + y′ + 4.25y = 0y(0) = 1y′(0) = 0
y(1) =

Answer key

  1. 1. m = 26.71 mg
  2. 2. m = 16.49 mg
  3. 3. y(0.5) = 1.151
  4. 4. y(2) = 0.04026
  5. 5. y(0.75) = −0.07764
  6. 6. y(1) = −0.1145

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