find y in terms of x given that x(dy/dx)+3y=e^x
And that when y=1 x=1
$\displaystyle x\frac{dy}{dx}+3y=e^x$
$\displaystyle \frac{dy}{dx}+3\frac{y}{x}=\frac{e^x}{x}$
Now multiply by the integrating factor $\displaystyle e^{\int \frac3{x} \, dx} = x^3$.
$\displaystyle \frac{d(yx^3)}{dx}=x^2 e^x$
$\displaystyle \int d(yx^3)=\int x^2 e^x \, dx$
$\displaystyle yx^3=(x^2 - 2x + 2)e^x + C$
$\displaystyle x\frac{dy}{dx} + 3y = e^x$
$\displaystyle \frac{dy}{dx} + \frac{3}{x}y = \frac{e^x}{x}$
Integrating factor = $\displaystyle e^{\int \frac{3}{x}\, dx}=e^{ln x^3} = x^3$
Multiply through $\displaystyle x^3$, giving $\displaystyle x^3\frac{dy}{dx} + 3x^2y = e^xx^2$
So $\displaystyle x^3 y = \int x^2e^x\, dx$
I would $\displaystyle \int x^2 e^x \, dx$ by parts:
$\displaystyle u=x^2, \frac{du}{dx} = 2x, \frac{dv}{dx} = e^x, v = e^x$
so $\displaystyle x^3y = e^xx^2 - \int 2xe^x\, dx$
$\displaystyle u=2x, \frac{du}{dx}=2, \frac{dv}{dx}=e^x, v=e^x$
$\displaystyle x^3y = e^xx^2 - 2xe^x + \int 2e^x \,dx = (x^2-2x+2)e^x + C$
To get $\displaystyle C$, substitute $\displaystyle y=1, x=1$ into $\displaystyle x^3y = (x^2-2x+2)e^x + C$
So $\displaystyle 1=(1-2+2)e + C => C = 1-e$
$\displaystyle x^3y=(x^2-2x+2)e^x + 1-e$