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 grows according to dy/dt = k·y with k > 0. It is 200 at t = 0 and 620 at t = 7 h. Find the rate constant k in h⁻¹.

y(0) = 200y(7) = 620

k = h⁻¹

2.Easy

A quantity grows according to dy/dt = k·y with k > 0. It is 120 at t = 0 and 300 at t = 7 h. Find the rate constant k in h⁻¹.

y(0) = 120y(7) = 300

k = h⁻¹

3.Medium

Solve y″ + 5y′ + 6y = 0 with y(0) = 2 and y′(0) = 2. Find y(2).

y″ + 5y′ + 6y = 0y(0) = 2y′(0) = 2

y(2) =

4.Medium

A can of soda at 3 °C is taken out of the fridge into a room at 22 °C. After 13 min its temperature is 17 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T₀ = 3 °CT_a = 22 °CT(13) = 17 °C

k = min⁻¹

5.Hard

A system obeys y″ + 2y′ + 17y = 17 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(0.5); angles in radians.

y″ + 2y′ + 17y = 17y(0) = 0y′(0) = 0

y(0.5) =

6.Hard

Solve y′ + 0.5y = 4e^−2t with y(0) = 0. Find y(1).

y′ + 0.5y = 4e^−2ty(0) = 0

y(1) =

Answer key

1. k = 0.1616 h⁻¹
2. k = 0.1309 h⁻¹
3. y(2) = 0.1317
4. k = 0.1027 min⁻¹
5. y(0.5) = 1.115
6. y(1) = 1.257

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