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 400 at t = 0 and 1040 at t = 10 h. Find the rate constant k in h⁻¹.

y(0) = 400y(10) = 1040

k = h⁻¹

2.Easy

A bacterial culture starts with 500 cells and grows according to dN/dt = k·N with k = 0.28 h⁻¹. After how many hours t are there 2500 cells?

N₀ = 500k = 0.28 h⁻¹N = 2500

t = h

3.Medium

A forged steel part at 380 °C is left to cool in a workshop at 29 °C. After 60 min its temperature is 46 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T₀ = 380 °CT_a = 29 °CT(60) = 46 °C

k = min⁻¹

4.Medium

A cup of coffee at 95 °C is left in a room at 20 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.09 min⁻¹, find its temperature T after 24 min.

T₀ = 95 °CT_a = 20 °Ck = 0.09 min⁻¹t = 24 min

T = °C

5.Hard

Solve y′ + y = 7e^−0.5t with y(0) = 3. Find y(0.5).

y′ + y = 7e^−0.5ty(0) = 3

y(0.5) =

6.Hard

Solve the initial value problem y′ + 3y/x = x for x > 0, with y(1) = 4. Find y(2).

y′ + 3y/x = xy(1) = 4

y(2) =

Answer key

1. k = 0.09555 h⁻¹
2. t = 5.748 h
3. k = 0.05046 min⁻¹
4. T = 28.65 °C
5. y(0.5) = 4.231
6. y(2) = 1.275

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