Solving a Nonlinear ODE: Seeking Guidance

[John Sebastia]

I was looking for some guidance on how to attack this problem.

Consider the nonlinear ODE:

y'(x)+y$$^{}2$$(x)+Ay(x)+B=0

(y prime + y squared with A and B constant coefficients)

Show that the solution is given by y=z'/z, where z(x) solves the second order ODE:

z''+Az'+Bz=0

(z double prime plus z prime with A and B constant coefficients)

Any advice would be greatly appreciated!

Thanks!

 

[tiny-tim]

Welcome to PF!

y'(x)+y$$^{}2$$(x)+Ay(x)+B=0
…
Show that the solution is given by y=z'/z, where z(x) solves the second order ODE:

z''+Az'+Bz=0

Hi John! Welcome to PF!

If y = z'/z, then y' = … ?

Now subsitute into the original equation, and multiply by z.

 

[John Sebastia]

Really. Is that all that needs to be done? I did that and got the original equation to look like the second one. So then that makes it a solution for the first equation right? Hmm, I guess that was easy. Thanks!