Question

Q:3 Verify that
$\frac{ \sin(x) }{ \sqrt{x} }$
is a solution of x^2y" + x y' + (x^2-1)/4 y = 0 over any interval on the positive axis and hence find its general solution.

Q:4 Solve the following differential equations :

a) y"-4y'+3y = 0



b) y" + 2y' + (w^2-1)y =0 where w is real.

Answer 1

1) Homogeneous Second Order Linear equations with constant Co -efficient s:

=>If ODE is of the form

ay"+by'+cy = 0
where x belongs to some interval.
and a, b, c are constants , then two. independent solutions (I. e. basis)
depend on the quadratic equation
am^2 + bm + c = 0

This equation is called Characteristic equation or Auxiliary equation.

2) Reduction of Order :

Consider the Homogeneous Second order equation

y" + p(x)y' + q(x)y = 0.


If we know one solution by any method, then it is easy to find other one by substituting y2= u y1 where u is some function.


Pic 1 : Verification in Q3
Pic 2 : GS of Q3
Pic 3 :GS OF Q:3
PIC 4 , pic 5: Solution of Q:4 a) and b)

Hope, you understand my answer and it may helps you.