Let the solution curve y=y(x) of (4+x^2),dy-2x(x^2+3y+4),dx=0 pass through the origin.

Question

Let the solution curve $y=y(x)$ of $(4+x^2)\,dy-2x(x^2+3y+4)\,dx=0$ pass through the origin. Then $y(2)$ equals \_\_\_\_\_\_.

Answer: 12

Accepted Answer

This is a linear ODE in $y$: $\dfrac{dy}{dx}-\dfrac{6x}{4+x^2}y = \dfrac{2x(x^2+4)}{4+x^2}=2x$; I.F. $=(4+x^2)^{-3}$. Solving with $y(0)=0$ and evaluating at $x=2$ gives $y(2)=12$.

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