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
ty1

A mass–spring–damper is described by y″ + 6y′ + 100y = 0. Compare it with y″ + 2ζωny′ + ωn²y = 0 and find the damping ratio ζ.

y″ + 6y′ + 100y = 0
ζ =
2.Easy
t (days)m (mg)34 dm = ?

A radioactive tracer has a half-life of 12 days. A sample initially contains 40 mg. The decay follows dm/dt = −k·m. How much m remains after 34 days?

m0 = 40 mgt½ = 12 dt = 34 d
m = mg
3.Medium
tyy(0) = 1y(1.5) = ?

Solve y″ + 8y′ + 15y = 0 with y(0) = 1 and y′(0) = -4. Find y(1.5).

y″ + 8y′ + 15y = 0y(0) = 1y′(0) = -4
y(1.5) =
4.Medium
tyy(0) = 1y(0.5) = ?

Solve y″ + 5y′ + 6y = 0 with y(0) = 1 and y′(0) = -4. Find y(0.5).

y″ + 5y′ + 6y = 0y(0) = 1y′(0) = -4
y(0.5) =
5.Hard
tyK = 2y(1.5) = ?

A system obeys y″ + y′ + 16.25y = 32.5 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(1.5); angles in radians.

y″ + y′ + 16.25y = 32.5y(0) = 0y′(0) = 0
y(1.5) =
6.Hard
t (min)m (kg)t = ?53 kg

A tank holds 500 L of water in which 5 kg of salt is dissolved. Brine with 0.25 kg of salt per litre flows in at 5 L/min, and the well-mixed solution flows out at the same rate. After how many minutes t does the tank contain 53 kg of salt?

V = 500 Lm0 = 5 kgc = 0.25 kg/Lr = 5 L/min
t = min

Answer key

  1. 1. ζ = 0.3
  2. 2. m = 5.612 mg
  3. 3. y(1.5) = 0.005831
  4. 4. y(0.5) = 0.07838
  5. 5. y(1.5) = 1.126
  6. 6. t = 51.08 min

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