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)38 dm = ?

A radioactive tracer has a half-life of 14 days. A sample initially contains 200 mg. The decay follows dm/dt = −k·m. How much m remains after 38 days?

m0 = 200 mgt½ = 14 dt = 38 d
m = mg
2.Easy
rr² + br + c?

For the equation y″ + 7y′ + 6y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 7y′ + 6y = 0
r =
3.Medium
tyy(0) = 1y(1.5) = ?

Solve y″ + 6y′ + 8y = 0 with y(0) = 1 and y′(0) = 0. Find y(1.5).

y″ + 6y′ + 8y = 0y(0) = 1y′(0) = 0
y(1.5) =
4.Medium
t (min)T (°C)7Ta​T = ?

A cup of coffee at 91 °C is left in a room at 19 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.12 min⁻¹, find its temperature T after 7 min.

T0 = 91 °CTa = 19 °Ck = 0.12 min⁻¹t = 7 min
T = °C
5.Hard
t (min)m (kg)t = ?100 kg

A tank holds 500 L of pure water. Brine with 0.25 kg of salt per litre flows in at 5 L/min, and the well-mixed solution flows out at the same rate. After how many minutes t does the tank contain 100 kg of salt?

V = 500 Lm0 = 0 kgc = 0.25 kg/Lr = 5 L/min
t = min
6.Hard
tyK = 1ymax​ = ?

A system obeys y″ + y′ + 25.25y = 25.25 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value ymax of its step response.

y″ + y′ + 25.25y = 25.25y(0) = 0y′(0) = 0
ymax =

Answer key

  1. 1. m = 30.48 mg
  2. 2. r = −1
  3. 3. y(1.5) = 0.0971
  4. 4. T = 50.08 °C
  5. 5. t = 160.9 min
  6. 6. ymax = 1.73

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