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)Nt = ?15000

A bacterial culture starts with 5000 cells and grows according to dN/dt = k·N with k = 0.13 h⁻¹. After how many hours t are there 15000 cells?

N0 = 5000k = 0.13 h⁻¹N = 15000
t = h
2.Easy
t (days)m (mg)51 dm = ?

A radioactive tracer has a half-life of 20 days. A sample initially contains 80 mg. The decay follows dm/dt = −k·m. How much m remains after 51 days?

m0 = 80 mgt½ = 20 dt = 51 d
m = mg
3.Medium
tyy(0) = 1y(1) = ?

Solve y″ + 2y′ + y = 0 with y(0) = 1 and y′(0) = 4. Find y(1).

y″ + 2y′ + y = 0y(0) = 1y′(0) = 4
y(1) =
4.Medium
t (min)T (°C)26Ta​T = ?

A can of soda at 3 °C is taken out of the fridge into a room at 22 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.03 min⁻¹, find its temperature T after 26 min.

T0 = 3 °CTa = 22 °Ck = 0.03 min⁻¹t = 26 min
T = °C
5.Hard
ty1.5y(1.5) = ?

Solve y′ + 0.5y = 8e−3t with y(0) = 0. Find y(1.5).

y′ + 0.5y = 8e−3ty(0) = 0
y(1.5) =
6.Hard
tyK = 1ymax​ = ?

A system obeys y″ + 4y′ + 13y = 13 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + 4y′ + 13y = 13y(0) = 0y′(0) = 0
ymax =

Answer key

  1. 1. t = 8.451 h
  2. 2. m = 13.66 mg
  3. 3. y(1) = 2.207
  4. 4. T = 13.29 °C
  5. 5. y(1.5) = 1.476
  6. 6. ymax = 1.123

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