If y=f(x) passes through (1,2) and satisfies xdydx+y=bx^4, for what b is _1^2 f(x),dx=625?

Question

If $y=f(x)$ passes through $(1,2)$ and satisfies $x\dfrac{dy}{dx}+y=bx^4$, for what $b$ is $\int_1^2 f(x)\,dx=\dfrac{62}{5}$?

Accepted Answer

I.F.$=x$; $y=\dfrac{bx^4}{5}+\dfrac{C}{x}$. Using $(1,2)$: $2=\dfrac b5+C$.

$$\int_1^2\left(\frac{bx^4}{5}+\frac{C}{x}\right)dx = \frac{62}{5}.$$

Solving this together with the point condition gives $b=10$.

Solutions · 1