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)3 dm = ?

A radioactive tracer has a half-life of 8 days. A sample initially contains 80 mg. The decay follows dm/dt = −k·m. How much m remains after 3 days?

m0 = 80 mgt½ = 8 dt = 3 d
m = mg
2.Easy
t (h)y12 h45

A quantity decays according to dy/dt = −k·y with k > 0. It is 100 at t = 0 and 45 at t = 12 h. Find the rate constant k in h⁻¹.

y(0) = 100y(12) = 45
k = h⁻¹
3.Medium
tyy(0) = 2y(0.5) = ?

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

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

Solve y″ + 4y′ + 3y = 0 with y(0) = 5 and y′(0) = 4. Find y(2).

y″ + 4y′ + 3y = 0y(0) = 5y′(0) = 4
y(2) =
5.Hard
tyK = 1ymax​ = ?

A system obeys y″ + 3y′ + 18.25y = 18.25 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + 3y′ + 18.25y = 18.25y(0) = 0y′(0) = 0
ymax =
6.Hard
tyK = 10y(0.25) = ?

A system obeys y″ + y′ + 25.25y = 252.5 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(0.25); angles in radians.

y″ + y′ + 25.25y = 252.5y(0) = 0y′(0) = 0
y(0.25) =

Answer key

  1. 1. m = 61.69 mg
  2. 2. k = 0.06654 h⁻¹
  3. 3. y(0.5) = 1.82
  4. 4. y(2) = 1.275
  5. 5. ymax = 1.308
  6. 6. y(0.25) = 6.38

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