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)31 dm = ?

A radioactive tracer has a half-life of 24 days. A sample initially contains 10 mg. The decay follows dm/dt = −k·m. How much m remains after 31 days?

m0 = 10 mgt½ = 24 dt = 31 d
m = mg
2.Easy
t (h)N21 hN = ?

A bacterial culture starts with 5000 cells and grows according to dN/dt = k·N with k = 0.22 h⁻¹. How many cells N are there after 21 h?

N0 = 5000k = 0.22 h⁻¹t = 21 h
N =
3.Medium
tyy(0) = 5y(0.5) = ?

Solve y″ + 6y′ + 8y = 0 with y(0) = 5 and y′(0) = -2. Find y(0.5).

y″ + 6y′ + 8y = 0y(0) = 5y′(0) = -2
y(0.5) =
4.Medium
tyy(0) = 4y(2) = ?

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

y″ + 4y′ + 3y = 0y(0) = 4y′(0) = 2
y(2) =
5.Hard
tyK = 1ymax​ = ?

A system obeys y″ + 4y′ + 29y = 29 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + 4y′ + 29y = 29y(0) = 0y′(0) = 0
ymax =
6.Hard
t (min)m (kg)90 minm = ?

A tank holds 1000 L of water in which 5 kg of salt is dissolved. Brine with 0.05 kg of salt per litre flows in at 20 L/min, and the well-mixed solution flows out at the same rate. How much salt m is in the tank after 90 min?

V = 1000 Lm0 = 5 kgc = 0.05 kg/Lr = 20 L/min
m = kg

Answer key

  1. 1. m = 4.085 mg
  2. 2. N = 507470
  3. 3. y(0.5) = 2.77
  4. 4. y(2) = 0.9399
  5. 5. ymax = 1.285
  6. 6. m = 42.56 kg

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