Serious problem about Differential equation

[kthouz]

Hi! I have 3 really problem here that i beg anyone who knows more to help me:
1. How to solve the equation of the kind xy'-3y=3 using power series.
I usually solve kind pf homogeneous equations. Here the problem rises when i want to equalise all the coefficient of x^n to 0. Because of that 3 there i don't now how to procede.
2. I don't know what is exactly an ordinary point. They say that an ordianry point is a point at which an ordinary differential equation is "analytic",but i don't understand that term "analytic".
3. Is it important to look for the radius of convergence first, when you want to solve an ODE using power series? I always consider around x=0.

 

[HallsofIvy]

Hi! I have 3 really problem here that i beg anyone who knows more to help me:
1. How to solve the equation of the kind xy'-3y=3 using power series.
I usually solve kind pf homogeneous equations. Here the problem rises when i want to equalise all the coefficient of x^n to 0. Because of that 3 there i don't now how to procede.

That is exactly why memorizing formulas without recognizing them is a mistake!

If this were xy'- 3y= 0,you would know what to do? Then you would write $y(x)= \sum_{n=0}^\infty a_n x^n$ so that $y'(x)= \sum_{n=1}^\infty n a_n x^{n-1}$ and your equation would be
$xy'- 3y= \sum_{n=1}^\infty n a_n x^n- \sum_{n=0}^\infty 3a_n x^n= 0$
Notice that the first sum is from 1 to infinity while the second sum is from 0 to infinity. That happens because when you differentiate a power series, you lose the first term. Of course, since you say you can solve homogenous equations, you know that you need to treat that case separately: If n= 0, 3a₀= 0 so a₀= 0. If n> 0 then
na_n- 3a_n= (n- 3)a_n= 0 which makes the solution kind of trivial! Either a_n= 0 or n= ? (Of course, it's a separable equation and easy to solve that way.)

The only difference now is that you have
$xy'- 3y= \sum_{n=1}^\infty n a_n x^n- \sum_{n=0}^\infty 3a_n x^n= 3$
You can think of that right hand side as 3x⁰ in order to match it to the left hand side. Again, do the n=0 case separately: 3a₀= 3 so a₀= 1 now. Do you see what to do if you had x² or 3+ x² on the right side? For some more complicated function, like e^x or cos(x), expand them in their Taylor series and then equate corresponding coefficients.

2. I don't know what is exactly an ordinary point. They say that an ordianry point is a point at which an ordinary differential equation is "analytic",but i don't understand that term "analytic".

A real valued function is analytic at a point if its Taylor's series exists at that point (so it must be infinitely differentiable in order to have a Taylor's series) and converges to the function on some neighborhood of the point. Caution- there exist functions that "have" Taylor's series but the radius of convergence is 0. There exist functions that have Taylor's series that converge for all x but not to the function! Fortunately for you almost all, if not all, of the examples you will see are "not analytic" because they have a factor in the denominator that goes to 0 at that point.

3. Is it important to look for the radius of convergence first, when you want to solve an ODE using power series? I always consider around x=0.

I wouldn't consider it "necessary", although it might save you some work if you found the radius of convergence was 0!

You shouldn't consider only "around x= 0". You might well have a problem where the "initial condition" is given at some point other than x= 0. If you were told that y(a)= 1, satisfying that condition will be much easier if you wrote $y= \Sum a_n (x- a)^n$. Or you could "shift" your variable: write t= x- a and rewrite the equation in terms of t rather than a.

 

[kthouz]

Thanks!
And can someone show me (give me hint on) how to find legendre polynomials on 1st and 2nd degrees using (1-2xh+h^2)^(-1/2)=∑h^n P_n(x) .