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)N11 hN = ?

A bacterial culture starts with 1000 cells and grows according to dN/dt = k·N with k = 0.24 h⁻¹. How many cells N are there after 11 h?

N0 = 1000k = 0.24 h⁻¹t = 11 h
N =
2.Easy
t (days)m (mg)83 dm = ?

A radioactive tracer has a half-life of 29 days. A sample initially contains 100 mg. The decay follows dm/dt = −k·m. How much m remains after 83 days?

m0 = 100 mgt½ = 29 dt = 83 d
m = mg
3.Medium
t (min)T (°C)33Ta​T = ?

A forged steel part at 300 °C is left to cool in a workshop at 26 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.01 min⁻¹, find its temperature T after 33 min.

T0 = 300 °CTa = 26 °Ck = 0.01 min⁻¹t = 33 min
T = °C
4.Medium
t (min)T (°C)26Ta​15 °C

A can of soda at 4 °C is taken out of the fridge into a room at 24 °C. After 26 min its temperature is 15 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 4 °CTa = 24 °CT(26) = 15 °C
k = min⁻¹
5.Hard
tyy(0) = 5y(1.5) = ?

Solve y″ + y′ + 25.25y = 0 with y(0) = 5 and y′(0) = 0 (released from rest). Find y(1.5); angles in radians.

y″ + y′ + 25.25y = 0y(0) = 5y′(0) = 0
y(1.5) =
6.Hard
tyK = 10ymax​ = ?

A system obeys y″ + y′ + 4.25y = 42.5 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + y′ + 4.25y = 42.5y(0) = 0y′(0) = 0
ymax =

Answer key

  1. 1. N = 14013
  2. 2. m = 13.75 mg
  3. 3. T = 223 °C
  4. 4. k = 0.03071 min⁻¹
  5. 5. y(1.5) = 1.04
  6. 6. ymax = 14.56

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