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Please provide an example!Why do we use \(\displaystyle \frac{y}{x}\) to get the general solution of a homogeneous differential equation? Why not \(\displaystyle \frac{x}{y}\)?

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If you show us example or problem you are trying to solve, we may help you..

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Suppose we have an ODE of the form:Why do we use \(\displaystyle \frac{y}{x}\) to get the general solution of a homogeneous differential equation? Why not \(\displaystyle \frac{x}{y}\)?

\(\displaystyle \d{y}{x}=f\left(\frac{y}{x}\right)\)

When we use the substitution:

\(\displaystyle u=\frac{y}{x}\implies y=ux\implies \d{y}{x}=u+x\d{u}{x}\)

We may now write the ODE as:

\(\displaystyle u+x\d{u}{x}=f(u)\)

We now have a separable equation:

\(\displaystyle \int\frac{1}{f(u)-u}\,du=\int \frac{1}{x}\,dx\)

Now suppose we choose to write:

\(\displaystyle \d{y}{x}=f\left(\frac{x}{y}\right)\)

\(\displaystyle u=\frac{x}{y}\implies y=\frac{x}{u}\implies \d{y}{x}=\frac{u-x\d{u}{x}}{u^2}\)

And our ODE becomes:

\(\displaystyle \frac{u-x\d{u}{x}}{u^2}=f(u)\)

\(\displaystyle x\d{u}{x}=u-u^2f(u)\)

\(\displaystyle \int \frac{1}{u-u^2f(u)}\,du=\int \frac{1}{x}\,dx\)

As we can see, both substitutions will lead to a separable ODE, but the first will likely be simpler. However, if you think that for a particular ODE the second substitution will more easily lead to a solution, then there's no reason you cannot use it.