Particular Solution of 2nd Order ODE: x^2y"-4xy'+6y=ln(x)

[november1992]

I was wondering what a guess would be for the particular solution of the right hand side of an equation if it looked like this:

x^{2}y" - 4xy' + 6y = ln(x)

My textbook has some specific examples of the right side function along with the corresponding form of the particular solution.

http://i.imgur.com/BT8AF.pngAlso, when is it best to use variation of parameters and the undetermined coefficients method to solve the equation?

 

[STEMucator]

You should really only use undetermined coefficients if the function you're solving for is either sin, cos, e, or a polynomial.

Variation of parameters could also be used, but generally undetermined coefficients is faster in those cases.

The thing is, for that equation, notice that you have a non-constant coefficient in front of your y'' term and perhaps using series would be a lot better for this equation...

 

[lurflurf]

rewrite this as
x^4 (y/x^2)''=log(x)

 

[Mentallic]

If you let x = e^t then the differential can be transformed into

z''-5z'+6z = t

It's called Euler's method I believe. Also notice the coefficient in front of the first derivative term has been reduced by 1.

 

[november1992]

You should really only use undetermined coefficients if the function you're solving for is either sin, cos, e, or a polynomial.

Variation of parameters could also be used, but generally undetermined coefficients is faster in those cases.

The thing is, for that equation, notice that you have a non-constant coefficient in front of your y'' term and perhaps using series would be a lot better for this equation...

That's what I assumed, but my teacher believes it's still faster to use variation of parameters. I don't like using that method because I tend to make more mistakes when I integrate e and a trig function.

If you let x = e^t then the differential can be transformed into

z''-5z'+6z = t

It's called Euler's method I believe. Also notice the coefficient in front of the first derivative term has been reduced by 1.

Would this method still be plausible if the right hand side was like this:

ln(x)*cos(3x)-3x^{2}

I was also wondering what to do in if I had a radical on the right hand side like this:

\sqrt{1-x^{2}}*e^{x}*sin(x)

 

[Mentallic]

Would this method still be plausible if the right hand side was like this:

ln(x)*cos(3x)-3x^{2}

I was also wondering what to do in if I had a radical on the right hand side like this:

\sqrt{1-x^{2}}*e^{x}*sin(x)

Yes the method still works, but finding your particular solution would be hard or it might not even have an exact algebraic solution.

But then again, you'd run into the same problem simply solving a second order linear ODE with those functions on the RHS.

 

[november1992]

I have another question. Can the undetermined coefficients method be used to solve cauchy-euler equations, or does variation of parameters have to be used?

 

[lurflurf]

^yes you can use undetermined coefficients
just work with x^a polynomials and trig functions of log(x) like
log(x)^2 sin(log(x) and x^17
cauchy-euler equations and linear constant coefficients are the same thing in different forms
try to solve
x²y" - 4xy' + 6y =(log(x))²sin(log(x))

with and without changing variables and see which you prefer, I think most people prefer to change it to linear constant coefficient form.

 

[november1992]

Oh, I actually meant could it be solved without changing the equation, but I guess that's a no.

 

[lurflurf]

It can be solved without changing, you just might not want to. I suggested that you do one both ways and compare. The two ways are exactly equivalent.