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 bacterial culture starts with 1000 cells and grows according to dN/dt = k·N with k = 0.1 h⁻¹. How many cells N are there after 4 h?

N₀ = 1000k = 0.1 h⁻¹t = 4 h

N =

2.Easy

A quantity decays according to dy/dt = −k·y with k > 0. It is 100 at t = 0 and 35 at t = 10 h. Find the rate constant k in h⁻¹.

y(0) = 100y(10) = 35

k = h⁻¹

3.Medium

Solve y″ + 4y′ + 4y = 0 with y(0) = 1 and y′(0) = -2. Find y(0.5).

y″ + 4y′ + 4y = 0y(0) = 1y′(0) = -2

y(0.5) =

4.Medium

A can of soda at 4 °C is taken out of the fridge into a room at 23 °C. After 15 min its temperature is 15 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T₀ = 4 °CT_a = 23 °CT(15) = 15 °C

k = min⁻¹

5.Hard

Solve the initial value problem y′ + 2y/x = 5x for x > 0, with y(1) = −1. Find y(3).

y′ + 2y/x = 5xy(1) = −1

y(3) =

6.Hard

Solve y′ + y = 6e^−0.5t with y(0) = 1. Find y(0.5).

y′ + y = 6e^−0.5ty(0) = 1

y(0.5) =

Answer key

1. N = 1492
2. k = 0.105 h⁻¹
3. y(0.5) = 0.3679
4. k = 0.05767 min⁻¹
5. y(3) = 11
6. y(0.5) = 2.674

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