Second order homogeneous linear differential equation

Question

The differential equation of the form

$x^2\frac{d^2y}{dx^2}+p_1\frac{dy}{dx}+p_2y=Q$ where $p_1$ ,$p_2$ are constants and Q the function of x is called Second order homogeneous linear equation.

How can we prove that above equation is second order homogeneous linear differential equation.And also help me to understand the significance of above equation in engineering field.

Answer 1

Just consider definitions and ask yourself questions:
1) Is it linear? Suppose you have two solutions
$y_{0},y_{1}$
Now add them together $a \cdot y_{0} + b\cdot y_{1}$
But the sum passes right through the derivatives to 0 for each solution
and the sum of zero's is zero.
That is the definition of linearity.
2) Is it homogeneous?
That requires that there all terms are scalar or derivative terms in y :
i.e $y,y',y'',y'''$
That normally means no free standing constant or $f(x)$ terms.

In all of Mathematics just read the definition and then read or think of examples. Then try to stretch the examples to limits and break the definition!
Sometimes this gets obscure and confusing, but this is at the heart of the structures that are being built. This distinguishing of mathematical objects that fall into one class; or out of it.
(Just a side note, find a variety of sources. Occasionally there are confusing typos even in textbooks. If you find differences, ask the teacher they are usually glad to get off the rote path and explain new things.)