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 2000 cells and grows according to dN/dt = k·N with k = 0.05 h⁻¹. How many cells N are there after 14 h?

N₀ = 2000k = 0.05 h⁻¹t = 14 h

N =

2.Easy

For the equation y″ + 3y′ + 2y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 3y′ + 2y = 0

r =

3.Medium

A forged steel part at 460 °C is left to cool in a workshop at 29 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.01 min⁻¹, find its temperature T after 27 min.

T₀ = 460 °CT_a = 29 °Ck = 0.01 min⁻¹t = 27 min

T = °C

4.Medium

A forged steel part at 320 °C is left to cool in a workshop at 20 °C. After 54 min its temperature is 195 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). Find the constant k in min⁻¹.

T₀ = 320 °CT_a = 20 °CT(54) = 195 °C

k = min⁻¹

5.Hard

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

y′ + 3y/x = 2y(1) = −2

y(3) =

6.Hard

Solve y″ + 4y′ + 20y = 0 with y(0) = 5 and y′(0) = 0 (released from rest). Find y(0.75); angles in radians.

y″ + 4y′ + 20y = 0y(0) = 5y′(0) = 0

y(0.75) =

Answer key

1. N = 4028
2. r = −1
3. T = 358 °C
4. k = 0.009981 min⁻¹
5. y(3) = 1.407
6. y(0.75) = −1.026

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