with work please. Use the reduction order method to find a second solution y2 vy

Question

with work please.

Use the reduction order method to find a second solution y2 vyito the differential equation knowing that the function y1(r) F is solution to that equation. a. (5/10) Find all functions v, that depend on a, such that v yi is solution of the differential equation in part (a). Denote by c and d the integration constants. Do not include d in your answer (ar) b. (5/10) Find a second solution y2 to the first differential equation above. Choose c and d such that the solution y2 does not contain any term proportional to y 1, and it satisfies the normalization y2(0) 1.

Explanation / Answer

Note that y = x is a solution to this differential equation.

I'll use Reduction of Order to find the general solution.

Let y = x * v(x) for some function u.
So, y' = v + xv' and y'' = 2v' + xv''.

Substituting into the DE yields
(x^2 - 1)(2v' + xv'') - 2x(v + xv') + 2xv = 0
==> x(x^2 - 1)v'' - 2v' = 0.

Separate variables:
v''/v' = 2/(x(x^2 - 1))
........= -2/x + 1/(x+1) + 1/(x-1) by partial fractions.

Integrate both sides:
ln(v') = -2 ln x + ln(x+1) + ln(x-1) + A
.........= ln ((x^2 - 1)/x^2) + A

So, v' = B(x^2 - 1)/x^2, by exponentiating both sides (and letting B = e^A).
...........= B(1 - 1/x^2).

Integrate both sides again:
v = C(x + 1/x) + d
==> y2 = xu = C(x^2 + 1) + dx.

y2(0)=1

==>1=C(0+1)+d*0

==>C=1

y2=1+x^2 as no proportional term to y1 was included

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