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
rr² + br + c?

For the equation y″ + 9y′ + 20y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 9y′ + 20y = 0
r =
2.Easy
t (days)m (mg)4 dm = ?

A radioactive tracer has a half-life of 14 days. A sample initially contains 10 mg. The decay follows dm/dt = −k·m. How much m remains after 4 days?

m0 = 10 mgt½ = 14 dt = 4 d
m = mg
3.Medium
tyy(0) = 5y(1) = ?

Solve y″ + 8y′ + 15y = 0 with y(0) = 5 and y′(0) = 0. Find y(1).

y″ + 8y′ + 15y = 0y(0) = 5y′(0) = 0
y(1) =
4.Medium
t (min)T (°C)25Ta​271 °C299 °C

A forged steel part at 340 °C is left to cool in a workshop at 30 °C. After 25 min its temperature is 271 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). How long after the start, t, does it take to reach 299 °C?

T0 = 340 °CTa = 30 °CT(25) = 271 °CT = 299 °C
t = min
5.Hard
tyy(0) = 3y(0.75) = ?

Solve y″ + 4y′ + 13y = 0 with y(0) = 3 and y′(0) = 0 (released from rest). Find y(0.75); angles in radians.

y″ + 4y′ + 13y = 0y(0) = 3y′(0) = 0
y(0.75) =
6.Hard
t (min)m (kg)120 minm = ?

A tank holds 400 L of pure water. Brine with 0.05 kg of salt per litre flows in at 5 L/min, and the well-mixed solution flows out at the same rate. How much salt m is in the tank after 120 min?

V = 400 Lm0 = 0 kgc = 0.05 kg/Lr = 5 L/min
m = kg

Answer key

  1. 1. r = −4
  2. 2. m = 8.203 mg
  3. 3. y(1) = 0.5718
  4. 4. t = 14.09 min
  5. 5. y(0.75) = −0.07327
  6. 6. m = 15.54 kg

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