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)N6 hN = ?

A bacterial culture starts with 1000 cells and grows according to dN/dt = k·N with k = 0.09 h⁻¹. How many cells N are there after 6 h?

N0 = 1000k = 0.09 h⁻¹t = 6 h
N =
2.Easy
t (days)m (mg)20 dm = ?

A radioactive tracer has a half-life of 11 days. A sample initially contains 100 mg. The decay follows dm/dt = −k·m. How much m remains after 20 days?

m0 = 100 mgt½ = 11 dt = 20 d
m = mg
3.Medium
t (min)T (°C)17Ta​T = ?

A can of soda at 5 °C is taken out of the fridge into a room at 27 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.1 min⁻¹, find its temperature T after 17 min.

T0 = 5 °CTa = 27 °Ck = 0.1 min⁻¹t = 17 min
T = °C
4.Medium
tyy(0) = 2y(1) = ?

Solve y″ + 8y′ + 16y = 0 with y(0) = 2 and y′(0) = -2. Find y(1).

y″ + 8y′ + 16y = 0y(0) = 2y′(0) = -2
y(1) =
5.Hard
tyy(0) = 10y(1) = ?

Solve y″ + 4y′ + 29y = 0 with y(0) = 10 and y′(0) = 0 (released from rest). Find y(1); angles in radians.

y″ + 4y′ + 29y = 0y(0) = 10y′(0) = 0
y(1) =
6.Hard
tyK = 10ymax​ = ?

A system obeys y″ + 3y′ + 6.25y = 62.5 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + 3y′ + 6.25y = 62.5y(0) = 0y′(0) = 0
ymax =

Answer key

  1. 1. N = 1716
  2. 2. m = 28.36 mg
  3. 3. T = 22.98 °C
  4. 4. y(1) = 0.1465
  5. 5. y(1) = −0.1352
  6. 6. ymax = 10.95

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