Which method to use?

1. Apr 7, 2005

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

2. Apr 7, 2005

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.

3. Apr 7, 2005

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?

4. Apr 7, 2005

dextercioby

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

Daniel.

5. Apr 7, 2005

estalniath

I see. Thank you for the advice =)

6. Apr 7, 2005