Question

(Higher-order linear differential equations) (a) Show that yi (x)-z?, уг (z)-r3, and U3(z) = 1/x are linearly independent solutions of 3. хзу",-z?y"-2xy' + 6y-0 on (-oo, 0) and (0, +00). Write down the general solution to (4 (b) Find a fundamental set S of solutions of

Answer 1

Given diff eq is

$x^3y'''-x^2y''-2xy'+6y=0$

y1(x) = x2 0-(2) 2r(2620 2 is solution of given diff eq

09-6-0 is solution of given diff eq.

1/x is solution of given diff eq.

$x^2,x^3,1/x$ are linearly independent :

$ax^2+bx^3+c1/x=0\\ ax^3+bx^4+c=0=0x^3+0x^4+0\\ \Rightarrow \\ a=b=c=0\\ \Rightarrow$

$x^2,x^3,1/x$ are linearly independent.

Hence $x^2,x^3,1/x$ are linearly independent solution of the given diff eq.

And general solution is

$y(x)=Ax^2+Bx^3+C/x$

(b)

write"d/dx"D uTL ar (D - 1)(D+1)(D 2) (D +3)0 D 1,,-1,2,-3