We've just started differential equations and the subsitution we use is y=vx, I was wondering whether thats the only substitution to use or whether there are others...and if there are others, how will I know which one to use?
Thanks
We've just started differential equations and the subsitution we use is y=vx, I was wondering whether thats the only substitution to use or whether there are others...and if there are others, how will I know which one to use?
Thanks
Either $\displaystyle y=ux,$ with $\displaystyle dy=u\,dx+x\,du,$ or $\displaystyle x=vy,$ with $\displaystyle dx=v\,dy+y\,dv,$ will reduce a first-order homogeneous equation to a separable one. Here's a quote from Zill, 6th Ed., p. 55:
Note that in Zill's notation, the DEAlthough either of the indicated substitutions can be used for every homogeneous differential equation, in practice we try $\displaystyle x=vy$ whenever the function $\displaystyle M(x,y)$ is simpler than $\displaystyle N(x,y)$.
$\displaystyle M(x,y)\,dx+N(x,y)\,dy=0$
is the homogeneous DE under discussion.
Does that answer your question?