# Homogeneous differential equation

#### burt

##### Junior Member
Why do we use $$\displaystyle \frac{y}{x}$$ to get the general solution of a homogeneous differential equation? Why not $$\displaystyle \frac{x}{y}$$?

#### Subhotosh Khan

##### Super Moderator
Staff member
Why do we use $$\displaystyle \frac{y}{x}$$ to get the general solution of a homogeneous differential equation? Why not $$\displaystyle \frac{x}{y}$$?

#### toponlinetool

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

#### MarkFL

##### Super Moderator
Staff member
Why do we use $$\displaystyle \frac{y}{x}$$ to get the general solution of a homogeneous differential equation? Why not $$\displaystyle \frac{x}{y}$$?
Suppose we have an ODE of the form:

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

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.