# Ordinary Differential Equations/Homogeneous x and y

Not to be confused with homogeneous equations, an equation homogeneous in x and y of degree n is an equation of the form$F(x,y,y')=0$Such that
$a^{n}F(x,y,y')=F(ax,ay,y')$.
Then the equation can take the form
$x^{n}F\left(1,{\frac {y}{x}},y'\right)=0$
Which is essentially another in the form
$x^{n}F\left({\frac {y}{x}},y'\right)=0$.
If we can solve this equation for $y'$, then we can easily use the substitution method mentioned earlier to solve this equation. Suppose, however, that it is more easily solved for ${\frac {y}{x}}$,
${\frac {y}{x}}=f(y')$
So that
$y=xf(y')$.
We can differentiate this to get
$y'=f(y')+xf'(y'){\frac {dy'}{dx}}$
Then re-arranging things,
${\frac {dx}{x}}={\frac {f'(y')}{y'-f(y')}}dy'$
So that upon integrating,
$ln(x)=\int {\frac {f'(y')}{y'-f(y')}}dy'+C$
We get
$x=Ce^{\int {\frac {f'(y')}{y'-f(y')}}dy'}$
Thus, if we can eliminate y' between two simultaneous equations
$y=xf(y')$
and
$x=Ce^{\int {\frac {f'(y')}{y'-f(y')}}dy'}$,
then we can obtain the general solution..

## Homogeneous Ordinary Differential Equations

A function P is homogeneous of order $\alpha$ if $a^{\alpha }P(x,y)=P(ax,ay)$. A homogeneous ordinary differential equation is an equation of the form P(x,y)dx+Q(x,y)dy=0 where P and Q are homogeneous of the same order.
The first usage of the following method for solving homogeneous ordinary differential equations was by Leibniz in 1691. Using the substitution y=vx or x=vy, we can make turn the equation into a separable equation.
    ${\frac {dy}{dx}}=F\left({\frac {y}{x}}\right)$
    $v(x,y)={\frac {y}{x}}$
    $y=vx\,$
Now we need to find v':
    ${\frac {dy}{dx}}=v+{\frac {dv}{dx}}x$
Plug back into the original equation
    $v+x{\frac {dv}{dx}}=F(v)$
    ${\frac {dv}{dx}}={\frac {F(v)-v}{x}}$
    Solve for v(x), then plug into the equation of v to get y
    $y(x)=xv(x)\,$
Again, don't memorize the equation. Remember the general method, and apply it.

### Example 2

    ${\frac {dy}{dx}}=5{\frac {y}{x}}+3{\frac {x}{y}}$
Let's use $v={\frac {y}{x}}$. Solve for $y'(x,v,v')$
    $y=vx\,$
    ${\frac {dy}{dx}}=v+x{\frac {dv}{dx}}$
Now plug into the original equation
    $v+x{\frac {dv}{dx}}=5v+{\frac {3}{v}}$
    $x{\frac {dv}{dx}}=4v+{\frac {3}{v}}$
    $v{\frac {dv}{dx}}={\frac {(4v^{2}+3)}{x}}$
Solve for v
    ${\frac {vdv}{4v^{2}+3}}={\frac {dx}{x}}$
    $\int {\frac {vdv}{4v^{2}+3}}=\int {\frac {dx}{x}}$
    ${\frac {1}{8}}\ln(4v^{2}+3)=\ln(x)$
    $4v^{2}+3=e^{8\ln(x)}\,$
    $4v^{2}+3=e^{\ln(x^{8})}\,$
    $4v^{2}+3=x^{8}\,$
    $v^{2}={\frac {x^{8}-3}{4}}$
Plug into the definition of v to get y.
    $y=vx\,$
    $y^{2}=v^{2}x^{2}\,$
    $y^{2}={\frac {x^{10}-3x^{2}}{4}}$
We leave it in $y^{2}$ form, since solving for y would lose information.
Note that there should be a constant of integration in the general solution. Adding it is left as an exercise.

### Example 3

    ${\frac {dy}{dx}}={\frac {x}{\sin({\frac {y}{x}})}}+{\frac {y}{x}}$
Lets use $v={\frac {y}{x}}$ again. Solve for $y'(x,v,v')$
    $y=vx\,$
    ${\frac {dy}{dx}}=v+x{\frac {dv}{dx}}$
Now plug into the original equation
    $v+x{\frac {dv}{dx}}={\frac {x}{\sin(v)}}+v$
    $x{\frac {dv}{dx}}={\frac {x}{\sin(v)}}$
    $\sin(v)dv=dx$
Solve for v:
    $\int \sin(v)dv=\int dx$
    $-\cos v=x+C\,$
    $v=\arccos(-x+C)\,$
Use the definition of v to solve for y.
    $y=vx\,$
    $y=\arccos(-x+C)x\,$

### An equation that is a function of a quotient of linear expressions

Given the equation $dy+f\left({\frac {a_{1}x+b_{1}y+c_{1}}{a_{2}x+b_{2}y+c_{2}}}\right)dx=0$,
We can make the substitution x=x'+h and y=y'+k where h and k satisfy the system of linear equations:
$a_{1}h+b_{1}k+c_{1}=0$ $a_{2}h+b_{2}k+c_{2}=0$
Which turns it into a homogeneous equation of degree 0:
$dy+f\left({\frac {a_{1}x'+b_{1}y'}{a_{2}x'+b_{2}y'}}\right)dx=0$