Consider the differential equation :
$y^{''} +py^{'}+qy=0$ where $p,q$ are continuous functions.
1.Show that $u_1(x)=x^a,u_2(x)=x^b$ are not solutions of the differential equation in any interval that contains the origin where $a,b>1$ are real.
2.Find $p,q$ such that $u_1,u_2$ are linearly independent solutions of the differential equation in some interval not containing the origin.
My try:
Since $u_1,u_2$ are solutions so they must satisfy the ODE:
$a(a-1)x^{a-2}+apx^{a-1}+qx^a=0$
$b(b-1)x^{b-2}+bpx^{b-1}+qx^b==0$
Multiplying the first equation by $x^b$ and second by $x^a$ we get the following:
$a(a-1)x^{a+b-2}+apx^{a+b-1}+qx^{a+b}=0$
$b(b-1)x^{a+b-2}+bpx^{a+b-1}+qx^{b+a}==0$
and solving we get:
$\dfrac{x^{a+b-2}}{ap-bq}=\dfrac{x^{a+b-1}}{b(b-1)q-qa(a-1)}=\dfrac{x^{a+b}}{a(a-1)bp-b(b-1)ap}$
But doing this also I cant prove that $u_1,u_2$ cant be solutions.
What should I do?
How should I find $p,q$?