Question

Solve : 9x(1-x)d2y/dx2-12dy/dx+4y=0​

Answer 1

Answer:

Solve in series the following ODE:

9x(1−x)y′′−12y′+4y=0

expanding y(x) about x0=0.

Step-by-step explanation:

4

Down vote

Accepted

You can check that x=0 and x=1 are singular regular points of the equation. Thus, you can obtain the solution in x∈(0,1) if you expand using Fröbenius about, for example, at x=0. If you do so, you can expand the solution as follows:

y(x)=∑n=0∞anxn+s,s∈C,a0≠0.

Plug this into the equation, provided uniform convergence of y,y′ and y′′, to have (double check please):

∑n=−1∞an+1((n+s+1)(9(n+s)−12))xn+s+∑n=0∞an(4−9(n+s)(n+s−1))=0(1)

Since the first series has one term more than second, we can get the term corresponding for n=−1 out of the series to come up with the known as indicial equation for s (if I did my maths well):

a0s(9(s−1)−12)=0⟹s=s1=0 or s=s2=1+12/9.

Since |s1−s2| is not a integer, we can conclude that there's no patology and Fröbenius will provide us the two parts of the solutions1, y1(x)=∑∞n=0an(s=s1)xn+s1 and y2(x)=∑∞n=0an(s=s2)xn+s2. The general term of an can be obtained from eq. (1) which for n≥0 reads:

an+1an=4−9(n+s)(n+s−1)(n+s+1)(9(n+s)−12).

It's up to you now to determine the value of an and check for convergence.

Cheers!

1, if this hadn't been the case, see here to see how you can obtain not only the particular solution (which in your case is 0) but the second part of the solution provided one of them.