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)60 dm = ?

A radioactive tracer has a half-life of 24 days. A sample initially contains 20 mg. The decay follows dm/dt = −k·m. How much m remains after 60 days?

m0 = 20 mgt½ = 24 dt = 60 d
m = mg
2.Easy
t (h)Nt = ?2000

A bacterial culture starts with 1000 cells and grows according to dN/dt = k·N with k = 0.15 h⁻¹. After how many hours t are there 2000 cells?

N0 = 1000k = 0.15 h⁻¹N = 2000
t = h
3.Medium
tyy(0) = 3y(0.5) = ?

Solve y″ + 4y′ + 4y = 0 with y(0) = 3 and y′(0) = -2. Find y(0.5).

y″ + 4y′ + 4y = 0y(0) = 3y′(0) = -2
y(0.5) =
4.Medium
tyy(0) = 1y(0.5) = ?

Solve y″ + 6y′ + 9y = 0 with y(0) = 1 and y′(0) = 2. Find y(0.5).

y″ + 6y′ + 9y = 0y(0) = 1y′(0) = 2
y(0.5) =
5.Hard
t (min)m (kg)45 minm = ?

A tank holds 500 L of water in which 10 kg of salt is dissolved. Brine with 0.2 kg of salt per litre flows in at 4 L/min, and the well-mixed solution flows out at the same rate. How much salt m is in the tank after 45 min?

V = 500 Lm0 = 10 kgc = 0.2 kg/Lr = 4 L/min
m = kg
6.Hard
tyK = 2ymax​ = ?

A system obeys y″ + 3y′ + 18.25y = 36.5 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + 3y′ + 18.25y = 36.5y(0) = 0y′(0) = 0
ymax =

Answer key

  1. 1. m = 3.536 mg
  2. 2. t = 4.621 h
  3. 3. y(0.5) = 1.839
  4. 4. y(0.5) = 0.781
  5. 5. m = 37.21 kg
  6. 6. ymax = 2.616

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