Solving a Differential Equation by Separation of Variables

[MidgetDwarf]

Solve the given differential equation by separation of variables.

(dy/dx)= (xy+3x-y-3)/(xy-2x+4y-8)

First, I noticed when i divided both sides by the left hand side and multiplied both sides by dx, nothing canceled or seemed to work.

I got to thinking.

on the right hand side I preformed long division.

i divided xy+3x-y-3 by xy-2x+4y-8.

I get 1 + (5x-5y+5)/(xy-2x+4y-8)

(dy/dx)= 1 + (5x-5y+5)/(xy-2x+4y-8)

I am stuck here. Any help is welcomed and appreciated.

 

[Tallus Bryne]

I think a better approach is beginning instead by factoring the RHS. From there it should be clear how to solve via separation of variables.

 

[MidgetDwarf]

I think a better approach is beginning instead by factoring the RHS. From there it should be clear how to solve via separation of variables.

wow, i over thought this problem. thanks a lot.

factoring the left hand side.

(y+3)(x-1)/(y-2)(x+4)

Left hand side= (y-2)/(y+3)dy

right hand side=(x-1)/(x+4)dx

then I integrate. the process is kind of lengthy, requiring trivial integration.

the answer is (x+4)^5=c(e^x)(e^-y)(y+3)^5.

I can't take you enough.

 

[Tallus Bryne]

the answer is (x+4)^5=c(e^x)(e^-y)(y+3)^5.

That looks like a correct implicit soln; you can also tidy up the RHS of your answer a little by applying some rules of exponents:
\begin{equation} e^{x}e^{-y} = e^{x-y} \end{equation}

 

[MidgetDwarf]

That looks like a correct implicit soln; you can also tidy up the RHS of your answer a little by applying some rules of exponents:
\begin{equation} e^{x}e^{-y} = e^{x-y} \end{equation}

yes, you are correct. thanks a lot.

Do you recommended a an intro ode book?

we are using zill in our class, and it is a bit to chatty. The graphics make the layout of the book a little hard to read in my opinion and he is too loose ( doesn't really use mathematical language) in his explanations.

 

[Tallus Bryne]

Do you recommended a an intro ode book?

I wouldn't be able to help you there. I also used a text co-authored by Zill when I took differential ('Differential Equations with Boundary-Value Problems' by Zill and Cullen 7ed.) when I took differential.
I saw a text by Ross recommended in the thread "How to self-[URL='https://www.physicsforums.com/insights/self-study-basic-high-school-mathematics/']study mathematics?[/URL]" However, I haven't ever had the opportunity to see what it's like myself.