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)70 dm = ?

A radioactive tracer has a half-life of 20 days. A sample initially contains 20 mg. The decay follows dm/dt = −k·m. How much m remains after 70 days?

m0 = 20 mgt½ = 20 dt = 70 d
m = mg
2.Easy
ty1

A mass–spring–damper is described by y″ + 19.2y′ + 64y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 19.2y′ + 64y = 0
ζ =
3.Medium
tyy(0) = 3y(2) = ?

Solve y″ + 8y′ + 16y = 0 with y(0) = 3 and y′(0) = 1. Find y(2).

y″ + 8y′ + 16y = 0y(0) = 3y′(0) = 1
y(2) =
4.Medium
t (min)T (°C)14Ta​T = ?

A cup of coffee at 94 °C is left in a room at 19 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.09 min⁻¹, find its temperature T after 14 min.

T0 = 94 °CTa = 19 °Ck = 0.09 min⁻¹t = 14 min
T = °C
5.Hard
tyK = 10y(0.75) = ?

A system obeys y″ + 2y′ + 5y = 50 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(0.75); angles in radians.

y″ + 2y′ + 5y = 50y(0) = 0y′(0) = 0
y(0.75) =
6.Hard
t (min)m (kg)t = ?30 kg

A tank holds 200 L of pure water. Brine with 0.25 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 30 kg of salt?

V = 200 Lm0 = 0 kgc = 0.25 kg/Lr = 2 L/min
t = min

Answer key

  1. 1. m = 1.768 mg
  2. 2. ζ = 1.2
  3. 3. y(2) = 0.009728
  4. 4. T = 40.27 °C
  5. 5. y(0.75) = 7.31
  6. 6. t = 91.63 min

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