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
rr² + br + c?

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

y″ + 5y′ + 6y = 0
r =
2.Easy
t (days)m (mg)7 dm = ?

A radioactive tracer has a half-life of 14 days. A sample initially contains 20 mg. The decay follows dm/dt = −k·m. How much m remains after 7 days?

m0 = 20 mgt½ = 14 dt = 7 d
m = mg
3.Medium
t (min)T (°C)28Ta​42 °C73 °C

A cup of coffee at 86 °C is left in a room at 21 °C. After 28 min its temperature is 42 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 73 °C?

T0 = 86 °CTa = 21 °CT(28) = 42 °CT = 73 °C
t = min
4.Medium
t (min)T (°C)18Ta​T = ?

A can of soda at 4 °C is taken out of the fridge into a room at 25 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.03 min⁻¹, find its temperature T after 18 min.

T0 = 4 °CTa = 25 °Ck = 0.03 min⁻¹t = 18 min
T = °C
5.Hard
tyK = 2y(0.5) = ?

A system obeys y″ + y′ + 4.25y = 8.5 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(0.5); angles in radians.

y″ + y′ + 4.25y = 8.5y(0) = 0y′(0) = 0
y(0.5) =
6.Hard
t (min)m (kg)t = ?17 kg

A tank holds 400 L of water in which 5 kg of salt is dissolved. Brine with 0.05 kg of salt per litre flows in at 4 L/min, and the well-mixed solution flows out at the same rate. After how many minutes t does the tank contain 17 kg of salt?

V = 400 Lm0 = 5 kgc = 0.05 kg/Lr = 4 L/min
t = min

Answer key

  1. 1. r = −2
  2. 2. m = 14.14 mg
  3. 3. t = 5.53 min
  4. 4. T = 12.76 °C
  5. 5. y(0.5) = 0.8308
  6. 6. t = 160.9 min

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