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)20 dm = ?

A radioactive tracer has a half-life of 7 days. A sample initially contains 20 mg. The decay follows dm/dt = −k·m. How much m remains after 20 days?

m0 = 20 mgt½ = 7 dt = 20 d
m = mg
2.Easy
rr² + br + c?

For the equation y″ + 6y′ + 5y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 6y′ + 5y = 0
r =
3.Medium
t (min)T (°C)6Ta​T = ?

A can of soda at 5 °C is taken out of the fridge into a room at 29 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.03 min⁻¹, find its temperature T after 6 min.

T0 = 5 °CTa = 29 °Ck = 0.03 min⁻¹t = 6 min
T = °C
4.Medium
tyy(0) = 5y(2) = ?

Solve y″ + 8y′ + 16y = 0 with y(0) = 5 and y′(0) = 4. Find y(2).

y″ + 8y′ + 16y = 0y(0) = 5y′(0) = 4
y(2) =
5.Hard
t (min)m (kg)t = ?84 kg

A tank holds 400 L of water in which 20 kg of salt is dissolved. Brine with 0.25 kg of salt per litre flows in at 10 L/min, and the well-mixed solution flows out at the same rate. After how many minutes t does the tank contain 84 kg of salt?

V = 400 Lm0 = 20 kgc = 0.25 kg/Lr = 10 L/min
t = min
6.Hard
ty1y(1) = ?

Solve y′ + y = 5e−3t with y(0) = 0. Find y(1).

y′ + y = 5e−3ty(0) = 0
y(1) =

Answer key

  1. 1. m = 2.76 mg
  2. 2. r = −1
  3. 3. T = 8.954 °C
  4. 4. y(2) = 0.01778
  5. 5. t = 64.38 min
  6. 6. y(1) = 0.7952

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