Let y=y(x) solve dydx+5x(x^5+1)y = (x^5+1)^2x^7, x0, y(1)=2.

Question

Let $y=y(x)$ solve $\dfrac{dy}{dx}+\dfrac{5}{x(x^5+1)}y = \dfrac{(x^5+1)^2}{x^7}$, $x>0$, $y(1)=2$. Then $y(2)$ is:

Accepted Answer

I.F. $= \dfrac{x^5}{1+x^5}$ (obtained via substitution $t=1+x^{-5}$).

$$y\cdot\frac{x^5}{1+x^5} = \int(x^3+x^{-2})\,dx = \frac{x^4}{4}-\frac1x+c.$$

Using $y(1)=2$: $1=\dfrac14-1+c \Rightarrow c=\dfrac74$.

At $x=2$: $y\cdot\dfrac{32}{33}=\dfrac{21}{4} \implies y=\dfrac{693}{128}$.

Solutions · 1