Verify that the function $y_1$ and $y_2$ are solutions of the given differential equation. Do they constitute a fundamental set of solutions?
$x^2y''-x(x+2)y' + (x+2)y = 0$, $x >0$, $y_1 = x$, $y_2 = xe^x$
$W(y_1, y_2)(x) = \det{\begin{vmatrix}y_1 & y_2\\
y_1' & y_2'\end{vmatrix}}
= \det{\begin{vmatrix}x & xe^x \\
1 & xe^x + e^x\end{vmatrix}}
= x(xe^x + e^x) - xe^x
= xe^x( x+1-1)
= x^2e^x $
since $x > 0$ and $e^x \neq 0$, so $W = x^2e^x \neq 0$.
Therefore, $y_1$ and $y_2$ constitute a fundamental set of solutions.
Verify :
1)Show $y_1 = x$ is one of the solutions
We know $y_1 = x$, $y_1' = 1$, $y_1'' = 0$, and then use these substitute in the equation, we have
LHS: $-x(x+2) + (x+2)x = 0$
RHS : 0
since LHS = RHS = 0, so $y_1 = x$ is one of the solutions
2) Show $y_2 = xe^x$ is one of the solutions
We know $y_2 = xe^x$, $y_2' = xe^x + e^x$, $y_2'' = e^x(x+2)$, also use thess substitute in the equation, we get
LHS: $x^2e^x(x+2)-x(x+2)(xe^x + e^x) + (x+2)xe^x = (x+2)e^x(x^2 -x(x+1) + x ) = (x+2)e^x(x^2 -x^2 - x + x ) = (x+2)e^x 0 = 0$
RHS : 0
since LHS = RHS = 0, so $y_2 = xe^x$ is one of the solutions.
