Nonlinear integration using integration factors?

[Kevatron9]

Hi,

I have some equations I have derived though energy balances and have been using the Integration Factors method to solve equations in the form:

dy/dx + Py = Q

However, my original equations only work if I assume some variables to be constants. Removing this assumption leaves me with an equation of the following form:

dy/dx + (a+b.y^2)y = Q

I have no Idea how to solve this problem. I cannot find a method that works. Does anybody know how to integrate such a problem?

Thanks in advance

 

[hunt_mat]

This looks like a Bernoulli type possibly.

 

[Kevatron9]

Thanks for the reply matt.

I have looked at the Bernoulli's method but as far as I am aware Bernoulli only works for linear integrations. Please advise me if I'm wrong and if possible, where I can see further information.

 

[RandomMystery]

Hi,

I have some equations I have derived though energy balances and have been using the Integration Factors method to solve equations in the form:

dy/dx + Py = Q

However, my original equations only work if I assume some variables to be constants. Removing this assumption leaves me with an equation of the following form:

dy/dx + (a+b.y^2)y = Q

I have no Idea how to solve this problem. I cannot find a method that works. Does anybody know how to integrate such a problem?

Thanks in advance

I'm not sure this is right, but it may work!:

dy/dx + (a+b.y^2)y = Q

dy/dx + ay+by^3 = Q

dy/dx = Q - ay - by^3

y = Qx - a$\int(ydx$-b$\int(y^3dx$

Now we convert dx into dy and y terms:

dy/dx = Q - ay - by^3

dx = $\frac{dy}{Q - ay - by^3}$

Bam! Here it is

y = Qx - a$\int(\frac{y}{Q - ay - by^3}dy$-b$\int(\frac{y^3}{Q - ay - by^3}dy$

 

[RandomMystery]

The above method is too tedious, here is a better way:

dy/dx = Q - ay - by^3

dx/dy = $\frac{1}{Q - ay - by^3}$

dx = $\frac{dy}{Q - ay - by^3}$

Now integrate both sides (I don't know how to do it) but Wolf math does:

http://integrals.wolfram.com/index.jsp?expr=1/(c+-+ax+-+bx^3)&random=false

Does anyone understand it's answer though?