Problem $3$:

Show that $y_1$ and $y_2$ are linearly independent and form a basis by taking their Wronskian

a$)y_1=e^{4x},y_2=e^{-1\mathrm{.}5x}$

b$)y_1=x,y_2=\frac{1}{x}$

c$)y_1=x^2,y_2=x^3$

d$)y_1=x^{k}cos(lnx),y_2=x^{k}sin(lnx)$