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

A quantity decays according to dy/dt = −k·y with k > 0. It is 200 at t = 0 and 60 at t = 2 h. Find the rate constant k in h⁻¹.

y(0) = 200y(2) = 60

k = h⁻¹

2.Easy

A bacterial culture starts with 1000 cells and grows according to dN/dt = k·N with k = 0.39 h⁻¹. How many cells N are there after 22 h?

N₀ = 1000k = 0.39 h⁻¹t = 22 h

N =

3.Medium

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

y″ + 4y′ + 4y = 0y(0) = 1y′(0) = -2

y(2) =

4.Medium

Solve y″ + 6y′ + 9y = 0 with y(0) = 1 and y′(0) = 4. Find y(1).

y″ + 6y′ + 9y = 0y(0) = 1y′(0) = 4

y(1) =

5.Hard

A system obeys y″ + 3y′ + 27.25y = 27.25 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(1.5); angles in radians.

y″ + 3y′ + 27.25y = 27.25y(0) = 0y′(0) = 0

y(1.5) =

6.Hard

A tank holds 200 L of water in which 10 kg of salt is dissolved. Brine with 0.25 kg of salt per litre flows in at 20 L/min, and the well-mixed solution flows out at the same rate. How much salt m is in the tank after 15 min?

V = 200 Lm₀ = 10 kgc = 0.25 kg/Lr = 20 L/min

m = kg

Answer key

1. k = 0.602 h⁻¹
2. N = 5.324 × 10⁶
3. y(2) = 0.01832
4. y(1) = 0.3983
5. y(1.5) = 0.9338
6. m = 41.07 kg

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