Substitution methods for first order differential equations

[eightlgddj]

Hi, I'm new to the forum, and new to differential equations. I was wondering if someone could post a no-nonsence explanation of substitution methods for first order differential equations.

Thanks!

 

[dextercioby]

I don't know what u may be speaking about.Are u referring to a separable diff.eq.?

$$a(x)\frac{dy}{dx}+b(x)f(y)=0$$

Daniel.

 

[eightlgddj]

Sorry if my post was confusing... I meant first order differential equations that are neither linear nor separable.

 

[dextercioby]

U mean something like that?

[atex] a(x)\left(\frac{dy}{dx}\right)^{k}+b(x)f(y)=c(x) [/tex]

The homegenous equation is separable.Therefore integrable.

Daniel.

 

[eightlgddj]

I mean one like this:

 

[dextercioby]

The homogenous equation is separable.

Daniel.

 

[eightlgddj]

I guess that's what I don't understand. How is it separable? I've been working on this for an hour, and I can't separate the variables. Could you show me how to work it out?

I really want to understand this, because I have to teach the concept of substitution methods to the rest of my class in a few weeks. My book says that this equation is not separable, and that the homogenous equation is only separable via substitution. And that's all it says.

 

[dextercioby]

The homogenous equation is

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

Since $y\neq 0$

,u get

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

,separate variables & integrate to get

$$y_{hom}(x)=Cx^{3/2}$$

Now,apply Lagrange's method to find the particular solution of the nonhomogenous one.

Daniel.

 

[eightlgddj]

I'm sorry, what's Lagrange's method? Its not in my book. I don't think its supposed to be that complicated of a solution, this is chapter 1 ODE stuff. There is no way to solve this problem by substitution?

 

[dextercioby]

Yes it can,make the substitution

$$y^{2}(x)=u(x)$$

Daniel.

 

[eightlgddj]

What happened to 4x^2?

 

[dextercioby]

It's there in the RHS,that substitution simplifies the integration of the ODE...



Daniel.

 

[saltydog]

I mean one like this:

That equation is homogeneous of degree 2. Thus, you can make a substitution y=vx, and then separate variables. Your ODE book should have this technique as a separate section at the beginning unless you have one of those "qualitative books" like Devaney's. I'm old-school.

 

[eightlgddj]

That equation is homogeneous of degree 2. Thus, you can make a substitution y=vx, and then separate variables. Your ODE book should have this technique as a separate section at the beginning unless you have one of those "qualitative books" like Devaney's. I'm old-school.

Thanks! Thats the substitution I came up with, I must have made a mistake after that point.