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 = ?2000

A bacterial culture starts with 200 cells and grows according to dN/dt = k·N with k = 0.26 h⁻¹. After how many hours t are there 2000 cells?

N0 = 200k = 0.26 h⁻¹N = 2000
t = h
2.Easy
t (days)m (mg)5 dm = ?

A radioactive tracer has a half-life of 3 days. A sample initially contains 20 mg. The decay follows dm/dt = −k·m. How much m remains after 5 days?

m0 = 20 mgt½ = 3 dt = 5 d
m = mg
3.Medium
t (min)T (°C)21Ta​T = ?

A can of soda at 5 °C is taken out of the fridge into a room at 28 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.04 min⁻¹, find its temperature T after 21 min.

T0 = 5 °CTa = 28 °Ck = 0.04 min⁻¹t = 21 min
T = °C
4.Medium
tyy(0) = 1y(1) = ?

Solve y″ + 7y′ + 12y = 0 with y(0) = 1 and y′(0) = -2. Find y(1).

y″ + 7y′ + 12y = 0y(0) = 1y′(0) = -2
y(1) =
5.Hard
xy12(1, 3)y(2) = ?

Solve the initial value problem y′ + y/x = 6x2 for x > 0, with y(1) = 3. Find y(2).

y′ + y/x = 6x2y(1) = 3
y(2) =
6.Hard
t (min)m (kg)75 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 10 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 = 1000 Lm0 = 5 kgc = 0.05 kg/Lr = 10 L/min
m = kg

Answer key

  1. 1. t = 8.856 h
  2. 2. m = 6.3 mg
  3. 3. T = 18.07 °C
  4. 4. y(1) = 0.08126
  5. 5. y(2) = 12.75
  6. 6. m = 28.74 kg

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