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 250 at t = 0 and 650 at t = 12 h. Find the rate constant k in h⁻¹.

y(0) = 250y(12) = 650

k = h⁻¹

2.Easy

A bacterial culture starts with 1500 cells and grows according to dN/dt = k·N with k = 0.17 h⁻¹. How many cells N are there after 10 h?

N₀ = 1500k = 0.17 h⁻¹t = 10 h

N =

3.Medium

A cup of coffee at 84 °C is left in a room at 21 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.1 min⁻¹, find its temperature T after 9 min.

T₀ = 84 °CT_a = 21 °Ck = 0.1 min⁻¹t = 9 min

T = °C

4.Medium

A forged steel part at 300 °C is left to cool in a workshop at 24 °C. After 57 min its temperature is 74 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T₀ = 300 °CT_a = 24 °CT(57) = 74 °C

k = min⁻¹

5.Hard

Solve y″ + 4y′ + 8y = 0 with y(0) = 5 and y′(0) = 0 (released from rest). Find y(0.25); angles in radians.

y″ + 4y′ + 8y = 0y(0) = 5y′(0) = 0

y(0.25) =

6.Hard

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

y′ + y/x = 2xy(1) = 4

y(2) =

Answer key

1. k = 0.07963 h⁻¹
2. N = 8211
3. T = 46.61 °C
4. k = 0.02997 min⁻¹
5. y(0.25) = 4.115
6. y(2) = 4.333

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