# Review:

## Separation of Variables

Some differential equations can be solved by separation of variables

### What is separation of variables?

All the y variables should be moved to one side of the equation, and all the x variables should be moved to the other side. |

### Process

- Separate the variables

- Integrate both sides

- Solve for y

- Substitute the initial condition to solve for C

## However, some of differential equations are too complicated that we cannot solve them by sepration of variables. Luckily, there is another way to solve these questions.

# What is homogeneous differential equation?

Some differential equations that are not separable in x and y can be made separable by a change of variables. This is true for differential equations of the form y'=f(x,y), ia homogeneous of degree n if

f(tx,ty)=t^{n}f(x,y)

Homogeneous means we can take a function: | f(x,y) |

Multiply each variable by t: | f(tx,ty) |

Then we can rearrange it to get this: | t^{n}f(x,y) |

So to be homogeneous a function must pass that test!

In other words: when each variable is multiplied by z, the result is mutiplied by some power of z:

f(tx,ty) = t^{n}f(x,y)

# How to varify homogeneous equations?

### Example:

start with | f(x,y) = x + 3y |

Multiply each variable by z: | f(zx,zy) = zx + 3zy |

Let's rearrange it by factoring out z: | f(zx,zy) = z(x + 3y) |

And x + 3y is f(x,y): | f(zx,zy) = zf(x,y) |

Which is what we wanted, with n=1: | f(zx,zy) = z^{1}f(x,y) |

# How to solve homogeneous differential equations?

We can solve it using Separation of Variables but first we create a new variable v=^{y}⁄_{x}

^{y}⁄

_{x}is also y = vx And

^{dy}⁄

_{dx}=

^{d(vx)}⁄

_{dx}=v

^{dx}⁄

_{dx}+x

^{dv}⁄

_{dx}(by the Product Rule) Which can be simplified to

^{dy}⁄

_{dx}=v+x

^{dv}⁄

_{dx}

using y=vx and ^{dy}⁄_{dx}=v+x^{dv}⁄_{dx} we can solve the Differential Equation.