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)23 dm = ?

A radioactive tracer has a half-life of 12 days. A sample initially contains 120 mg. The decay follows dm/dt = −k·m. How much m remains after 23 days?

m0 = 120 mgt½ = 12 dt = 23 d
m = mg
2.Easy
rr² + br + c?

For the equation y″ + 7y′ + 12y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 7y′ + 12y = 0
r =
3.Medium
t (min)T (°C)36Ta​127 °C

A forged steel part at 320 °C is left to cool in a workshop at 28 °C. After 36 min its temperature is 127 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 320 °CTa = 28 °CT(36) = 127 °C
k = min⁻¹
4.Medium
t (min)T (°C)18Ta​33 °C52 °C

A cup of coffee at 81 °C is left in a room at 23 °C. After 18 min its temperature is 33 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 52 °C?

T0 = 81 °CTa = 23 °CT(18) = 33 °CT = 52 °C
t = min
5.Hard
t (min)m (kg)40 minm = ?

A tank holds 1000 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. How much salt m is in the tank after 40 min?

V = 1000 Lm0 = 0 kgc = 0.25 kg/Lr = 2 L/min
m = kg
6.Hard
tyK = 1y(0.75) = ?

A system obeys y″ + 4y′ + 8y = 8 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(0.75); angles in radians.

y″ + 4y′ + 8y = 8y(0) = 0y′(0) = 0
y(0.75) =

Answer key

  1. 1. m = 31.78 mg
  2. 2. r = −3
  3. 3. k = 0.03005 min⁻¹
  4. 4. t = 7.098 min
  5. 5. m = 19.22 kg
  6. 6. y(0.75) = 0.7616

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