Question

If for \(x \ge 0\), \(y = y(x)\) is the solution of the differential equation \((x+1)\,dy = \left((x+1)^2 + y - 3\right)dx\), \(y(2) = 0\), then \(y(3)\) is equal to ______.

Hint:

Rewrite as a linear equation \(\frac{dy}{dx} - \frac{y}{x+1} = (x+1) - \frac{3}{x+1}\) and use the integrating factor \(\frac{1}{x+1}\).

Answer 1

Correct Answer: 3

\(\frac{dy}{dx} - \frac{y}{x+1} = (x+1) - \frac{3}{x+1}\).
Integrating factor \(= e^{-\int \frac{dx}{x+1}} = \frac{1}{x+1}\).
\(\frac{d}{dx}\left(\frac{y}{x+1}\right) = 1 - \frac{3}{(x+1)^2}\), so \(\frac{y}{x+1} = x + \frac{3}{x+1} + C\).
Using \(y(2) = 0\): \(0 = 2 + 1 + C \Rightarrow C = -3\).
At \(x=3\): \(\frac{y}{4} = 3 + \frac{3}{4} - 3 = \frac34 \Rightarrow y = 3\).
Answer: 3