Question

Suppose y₁ and y₂ are two solutions of the equation t²y^''+2t²y^'-t^-2y=0. Find W(y₁.y₂)(t).

Answer 1

Abel’s Theorem:

If $y_1$ and $y_2$ are any two solutions of the equation 

$$y''+p(t)y'+q(t)y=0$$

where $p$ and $q$ are continuous on an open interval $I$. Then the Wronskian $W(y_1, y_2)(t)$ is given by 

$$W(y_1, y_2)(t)=Ce^{-\int p(t)dt}$$

where C is a constant that depends on $y_1$ and $y_2$ but not on $t.$

$$y''+2y'-t^{-4}y=0$$

$p(t)=2$

$$W(y_1, y_2)(t)=Ce^{-\int 2dt}$$

$$W(y_1, y_2)(t)=Ce^{-2t}$$