# Thread: What types of ODE are these?

1. ## What types of ODE are these?

$(1+e^x)y dy +2(x+e^x)dy=0$

$xy^3dy-(x^4+y^4)dx=0$

The first one I am not sure at all about but is the second one homogenous with a substitution required of $v=\frac{y}{x}$ ?

2. ## Re: What types of ODE are these?

Originally Posted by Paulo1913
$(1+e^x)y dy +2(x+e^x)dy=0$

$xy^3dy-(x^4+y^4)dx=0$

The first one I am not sure at all about but is the second one homogenous with a substitution required of $v=\frac{y}{x}$ ?
For the first one I guess that there is a typo . Second one is non-exact differential equation so you should apply Integrating Factor Technique :

Let's denote :

$M=-(x^4+y^4) ~\text{and}~N=xy^3$

$\frac{\partial M}{\partial y}=-4y^3 ~ \text{and}~\frac{\partial N}{\partial x}=y^3$

Since :

$\frac{\frac{\partial M}{\partial y}-\frac{\partial N}{\partial x}}{N}=\frac{-5y^3}{xy^3}=\frac{-5}{x}$

Integrating actor is :

$u(x)=e^{-\int \frac{5}{x} \,dx}$

3. ## Re: What types of ODE are these?

Oh I see, thanks. For the first one, it should be a dx for the first term, not another dy.