Answer: y(x) = [tex]C_{1} e^{x} + C_{2} e^{\frac{-x}{2} }[/tex]
Step-by-step explanation:
2y ′′ − y ′ − y = 0
The characteristic equation is:
[tex]2r^{2} - r - 1 = 0[/tex]
[tex]2r^{2} - 2r + r - 1 = 0[/tex]
2r(r-1) + 1(r-1) = 0
(r-1)(2r+1) = 0
[tex]r_{1} = 1 , r_{2} = \frac{-1}{2}[/tex]
∴ there are two distinct roots
so the general equation is as follows:
y(x) = [tex]C_{1} e^{r_{1}x } + C_{2} e^{r_{2}x }[/tex]
y(x) = [tex]C_{1} e^{x} + C_{2} e^{\frac{-x}{2} }[/tex]
