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)71 dm = ?

A radioactive tracer has a half-life of 25 days. A sample initially contains 200 mg. The decay follows dm/dt = −k·m. How much m remains after 71 days?

m0 = 200 mgt½ = 25 dt = 71 d
m = mg
2.Easy
t (days)m (mg)31 dm = ?

A radioactive tracer has a half-life of 14 days. A sample initially contains 10 mg. The decay follows dm/dt = −k·m. How much m remains after 31 days?

m0 = 10 mgt½ = 14 dt = 31 d
m = mg
3.Medium
tyy(0) = 5y(0.5) = ?

Solve y″ + 6y′ + 8y = 0 with y(0) = 5 and y′(0) = -2. Find y(0.5).

y″ + 6y′ + 8y = 0y(0) = 5y′(0) = -2
y(0.5) =
4.Medium
tyy(0) = 2y(2) = ?

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

y″ + 2y′ + y = 0y(0) = 2y′(0) = 0
y(2) =
5.Hard
tyK = 1ymax​ = ?

A system obeys y″ + 2y′ + 26y = 26 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + 2y′ + 26y = 26y(0) = 0y′(0) = 0
ymax =
6.Hard
tyy(0) = 3y(0.75) = ?

Solve y″ + 3y′ + 6.25y = 0 with y(0) = 3 and y′(0) = 0 (released from rest). Find y(0.75); angles in radians.

y″ + 3y′ + 6.25y = 0y(0) = 3y′(0) = 0
y(0.75) =

Answer key

  1. 1. m = 27.93 mg
  2. 2. m = 2.155 mg
  3. 3. y(0.5) = 2.77
  4. 4. y(2) = 0.812
  5. 5. ymax = 1.533
  6. 6. y(0.75) = 0.7975

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