Help rearranging a linear first order differential equation

[Kris1]

Hi I am trying to solve dy/dx = 3x^2-2x+2+(8/x *y)

Can anyone show me how to rearrange to standard form as I am mightly confused :(

 

[Kris1]

Re: Help rearranging a linear first order differential

The equation of interest is the first of the two you have listed. Thankyou :)

 

[Chris L T521]

Re: Help rearranging a linear first order differential

Hi I am trying to solve dy/dx = 3x^2-2x+2+(8/x *y)

Can anyone show me how to rearrange to standard form as I am mightly confused :(

The first thing you need to do is get every term involving y on one side of the equation. So subtracting $\dfrac{8}{x}y$ from both sides gives you
\[\frac{dy}{dx}-\frac{8}{x}y=3x^2-2x+2.\]
We now note that the equation is now in the form of a linear equation $\dfrac{dy}{dx}+P(x)y=Q(x)$. To proceed from here, you need to compute the integrating factor
\[\mu(x)=\exp\left(\int P(x)\,dx\right)=\ldots\quad(\text{I leave this part to you})\]
where $\exp(x)=e^x$. Then if you multiply both sides of the linear ODE by $\mu(x)$, you get
\[\frac{d}{dx}[\mu(x) y]=\mu(x)(3x^2-2x+2)\implies y=\frac{1}{\mu(x)}\int \mu(x)(3x^2-2x+2)\,dx.\]

Can you fill in the work I left out? I hope this helps!

 

[Kris1]

Re: Help rearranging a linear first order differential

Yes thanks I can fill the rest out I was just unsure as how to rearrange the equation because of all the terms multiplied by x but I see that it is quite easy now :)