The differential equation $\frac{dy}{dx} = \frac{(x + y + 2)^2 - (y - 1)^2}{(x + 3)^2}$ determines a curve $y = f(x)$ & $f(0) = 1$ then $f(3)$ is equal to ____.
Step-by-Step Solution
Key Concept: Recognize the numerator $(x + y + 2)^2 - (y - 1)^2$ as a difference of squares that factors to $(x + 3)(x + 2y + 1)$, then use the substitution $x + 3 = X$ and $y - 1 = Y$ to convert the equation into a linear first-order ODE of the form $ rac{dY}{dX} - rac{2Y}{X} = 1$ solvable via integrating factor method.
Using substitution $x + 3 = X$ and $y - 1 = Y$, the equation becomes $\frac{dY}{dX} = \frac{X^2 + 2XY}{X^2}$. Rearranging: $\frac{dY}{dX} - \frac{2Y}{X} = 1$. The integrating factor is $e^{\int -\frac{2}{X}dx} = e^{-2\ln X} = \frac{1}{X^2}$. Multiplying and integrating: $Y \cdot \frac{1}{X^2} = \int \frac{1}{X^2} dx + c$, yielding the solution in terms of the original variables.
Correct Answer: 7