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)N10 hN = ?

A bacterial culture starts with 500 cells and grows according to dN/dt = k·N with k = 0.31 h⁻¹. How many cells N are there after 10 h?

N0 = 500k = 0.31 h⁻¹t = 10 h
N =
2.Easy
t (days)m (mg)79 dm = ?

A radioactive tracer has a half-life of 30 days. A sample initially contains 20 mg. The decay follows dm/dt = −k·m. How much m remains after 79 days?

m0 = 20 mgt½ = 30 dt = 79 d
m = mg
3.Medium
tyy(0) = 1y(0.5) = ?

Solve y″ + 5y′ + 6y = 0 with y(0) = 1 and y′(0) = -4. Find y(0.5).

y″ + 5y′ + 6y = 0y(0) = 1y′(0) = -4
y(0.5) =
4.Medium
tyy(0) = 3y(1.5) = ?

Solve y″ + 6y′ + 8y = 0 with y(0) = 3 and y′(0) = 6. Find y(1.5).

y″ + 6y′ + 8y = 0y(0) = 3y′(0) = 6
y(1.5) =
5.Hard
t (min)m (kg)75 minm = ?

A tank holds 200 L of water in which 10 kg of salt is dissolved. Brine with 0.25 kg of salt per litre flows in at 5 L/min, and the well-mixed solution flows out at the same rate. How much salt m is in the tank after 75 min?

V = 200 Lm0 = 10 kgc = 0.25 kg/Lr = 5 L/min
m = kg
6.Hard
tyK = 1ymax​ = ?

A system obeys y″ + 4y′ + 8y = 8 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + 4y′ + 8y = 8y(0) = 0y′(0) = 0
ymax =

Answer key

  1. 1. N = 11099
  2. 2. m = 3.223 mg
  3. 3. y(0.5) = 0.07838
  4. 4. y(1.5) = 0.4332
  5. 5. m = 43.87 kg
  6. 6. ymax = 1.043

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