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)y6 h296

A quantity grows according to dy/dt = k·y with k > 0. It is 80 at t = 0 and 296 at t = 6 h. Find the rate constant k in h⁻¹.

y(0) = 80y(6) = 296
k = h⁻¹
2.Easy
t (days)m (mg)8 dm = ?

A radioactive tracer has a half-life of 10 days. A sample initially contains 80 mg. The decay follows dm/dt = −k·m. How much m remains after 8 days?

m0 = 80 mgt½ = 10 dt = 8 d
m = mg
3.Medium
tyy(0) = 3y(1.5) = ?

Solve y″ + 2y′ + y = 0 with y(0) = 3 and y′(0) = 4. Find y(1.5).

y″ + 2y′ + y = 0y(0) = 3y′(0) = 4
y(1.5) =
4.Medium
t (min)T (°C)11Ta​T = ?

A cup of coffee at 85 °C is left in a room at 23 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.08 min⁻¹, find its temperature T after 11 min.

T0 = 85 °CTa = 23 °Ck = 0.08 min⁻¹t = 11 min
T = °C
5.Hard
t (min)m (kg)t = ?31 kg

A tank holds 400 L of water in which 10 kg of salt is dissolved. Brine with 0.2 kg of salt per litre flows in at 2 L/min, and the well-mixed solution flows out at the same rate. After how many minutes t does the tank contain 31 kg of salt?

V = 400 Lm0 = 10 kgc = 0.2 kg/Lr = 2 L/min
t = min
6.Hard
xy13(1, 0)y(3) = ?

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

y′ + 3y/x = 2x2y(1) = 0
y(3) =

Answer key

  1. 1. k = 0.2181 h⁻¹
  2. 2. m = 45.95 mg
  3. 3. y(1.5) = 3.012
  4. 4. T = 48.72 °C
  5. 5. t = 71.33 min
  6. 6. y(3) = 8.988

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