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)35 dm = ?

A radioactive tracer has a half-life of 19 days. A sample initially contains 200 mg. The decay follows dm/dt = −k·m. How much m remains after 35 days?

m0 = 200 mgt½ = 19 dt = 35 d
m = mg
2.Easy
rr² + br + c?

For the equation y″ + 5y′ + 4y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 5y′ + 4y = 0
r =
3.Medium
t (min)T (°C)22Ta​37 °C

A cup of coffee at 81 °C is left in a room at 21 °C. After 22 min its temperature is 37 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 81 °CTa = 21 °CT(22) = 37 °C
k = min⁻¹
4.Medium
tyy(0) = 1y(2) = ?

Solve y″ + 6y′ + 8y = 0 with y(0) = 1 and y′(0) = 4. Find y(2).

y″ + 6y′ + 8y = 0y(0) = 1y′(0) = 4
y(2) =
5.Hard
tyK = 1y(1.5) = ?

A system obeys y″ + 3y′ + 6.25y = 6.25 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(1.5); angles in radians.

y″ + 3y′ + 6.25y = 6.25y(0) = 0y′(0) = 0
y(1.5) =
6.Hard
t (min)m (kg)t = ?25 kg

A tank holds 500 L of pure water. Brine with 0.1 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 25 kg of salt?

V = 500 Lm0 = 0 kgc = 0.1 kg/Lr = 4 L/min
t = min

Answer key

  1. 1. m = 55.78 mg
  2. 2. r = −1
  3. 3. k = 0.06008 min⁻¹
  4. 4. y(2) = 0.07226
  5. 5. y(1.5) = 1.093
  6. 6. t = 86.64 min

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