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 (h)y2 h175

A quantity decays according to dy/dt = −k·y with k > 0. It is 500 at t = 0 and 175 at t = 2 h. Find the rate constant k in h⁻¹.

y(0) = 500y(2) = 175
k = h⁻¹
2.Easy
ty1

A mass–spring–damper is described by y″ + 6y′ + 36y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 6y′ + 36y = 0
ζ =
3.Medium
t (min)T (°C)41Ta​299 °C

A forged steel part at 440 °C is left to cool in a workshop at 20 °C. After 41 min its temperature is 299 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T0 = 440 °CTa = 20 °CT(41) = 299 °C
k = min⁻¹
4.Medium
tyy(0) = 2y(2) = ?

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

y″ + 4y′ + 4y = 0y(0) = 2y′(0) = 0
y(2) =
5.Hard
ty2y(2) = ?

Solve y′ + y = 9e−2t with y(0) = 2. Find y(2).

y′ + y = 9e−2ty(0) = 2
y(2) =
6.Hard
t (min)m (kg)40 minm = ?

A tank holds 200 L of pure water. Brine with 0.25 kg of salt per litre flows in at 10 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 = 200 Lm0 = 0 kgc = 0.25 kg/Lr = 10 L/min
m = kg

Answer key

  1. 1. k = 0.5249 h⁻¹
  2. 2. ζ = 0.5
  3. 3. k = 0.009977 min⁻¹
  4. 4. y(2) = 0.1832
  5. 5. y(2) = 1.324
  6. 6. m = 43.23 kg

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