Solve Differential Eq.: dy/dx= 1/(xy+x^2y^3)

[estalniath]

Hello guys,

Which method can we use to solve the differential equation below?
dy/dx= 1/(xy+x^2y^3)

It doesn't seem to be of any form which I had studied before(linear differential equations,bernoulli,exact differential,homogeneous,seperable equations) for first order differential equations yet it appeared in one of out past examination questions

 

[dextercioby]

It's easy.For me

$$\frac{dy}{dx}=\frac{1}{xy+x^{2}y^{3}}$$

Therefore

$$\frac{dx}{dy}=yx+y^{3}x^{2}$$

Make the substitution

$$x=\frac{1}{u}$$

,under which

$$\frac{dx}{dy}=-\frac{1}{u^{2}}\frac{du}{dy}$$

Therefore the new ODE is

$$-\frac{1}{u^{2}}\frac{du}{dy}=\frac{y}{u}+y^{3}\frac{1}{u^{2}}$$

equivalently

$$\frac{du}{dy}=-yu-y^{3}$$

I trust you can take it from here.It's an nonhomegenous linear ODE...(the homogenous eq is separable).


Daniel.

 

[estalniath]

Wow! That was a fast reply. =) Thank you for replying and how do you type the mathematical equations in between? I don't seem to see any functions in this thread which allows us to type mathematical functions?

 

[dextercioby]

Write formulas in latex code and use preview option for checking it b4 clicking submit.

Daniel.

 

[estalniath]

I see. Thank you for the advice =)

 

[Lyuokdea]

Here's a link to all the Latex info you need:

https://www.physicsforums.com/showthread.php?t=8997&highlight=Latex

~Lyuokdea