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 875 at t = 12 h. Find the rate constant k in h⁻¹.

y(0) = 250y(12) = 875

k = h⁻¹

2.Easy

For the equation y″ + 5y′ + 4y = 0, find the root r of the characteristic equation that is closest to zero. It sets how slowly the solution dies out.

y″ + 5y′ + 4y = 0

r =

3.Medium

A cup of coffee at 90 °C is left in a room at 20 °C. It follows Newton's law of cooling, dT/dt = −k·(T − Tₐ). With k = 0.11 min⁻¹, find its temperature T after 28 min.

T₀ = 90 °CT_a = 20 °Ck = 0.11 min⁻¹t = 28 min

T = °C

4.Medium

Solve y″ + 6y′ + 5y = 0 with y(0) = 1 and y′(0) = 4. Find y(2).

y″ + 6y′ + 5y = 0y(0) = 1y′(0) = 4

y(2) =

5.Hard

A system obeys y″ + 4y′ + 8y = 40 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(0.5); angles in radians.

y″ + 4y′ + 8y = 40y(0) = 0y′(0) = 0

y(0.5) =

6.Hard

A system obeys y″ + 2y′ + 10y = 20 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find y(0.5); angles in radians.

y″ + 2y′ + 10y = 20y(0) = 0y′(0) = 0

y(0.5) =

Answer key

1. k = 0.1044 h⁻¹
2. r = −1
3. T = 23.22 °C
4. y(2) = 0.3044
5. y(0.5) = 2.458
6. y(0.5) = 1.511

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