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)13 dm = ?

A radioactive tracer has a half-life of 19 days. A sample initially contains 40 mg. The decay follows dm/dt = −k·m. How much m remains after 13 days?

m0 = 40 mgt½ = 19 dt = 13 d
m = mg
2.Easy
t (h)N22 hN = ?

A bacterial culture starts with 1500 cells and grows according to dN/dt = k·N with k = 0.03 h⁻¹. How many cells N are there after 22 h?

N0 = 1500k = 0.03 h⁻¹t = 22 h
N =
3.Medium
tyy(0) = 5y(1) = ?

Solve y″ + 2y′ + y = 0 with y(0) = 5 and y′(0) = 0. Find y(1).

y″ + 2y′ + y = 0y(0) = 5y′(0) = 0
y(1) =
4.Medium
tyy(0) = 3y(2) = ?

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

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

A system obeys y″ + 2y′ + 10y = 50 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + 2y′ + 10y = 50y(0) = 0y′(0) = 0
ymax =
6.Hard
xy13(1, 0)y(3) = ?

Solve the initial value problem y′ + y/x = 6x for x > 0, with y(1) = 0. Find y(3).

y′ + y/x = 6xy(1) = 0
y(3) =

Answer key

  1. 1. m = 24.89 mg
  2. 2. N = 2902
  3. 3. y(1) = 3.679
  4. 4. y(2) = 1.759
  5. 5. ymax = 6.755
  6. 6. y(3) = 17.33

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