Question

Find the general solution to 2y ′′ − y ′ − y = 0.

Answer 1

Answer: y(x) = $C_{1} e^{x} + C_{2} e^{\frac{-x}{2} }$

Step-by-step explanation:

2y ′′ − y ′ − y = 0

The characteristic equation is:

$2r^{2} - r - 1 = 0$

$2r^{2} - 2r + r - 1 = 0$

2r(r-1) + 1(r-1) = 0

(r-1)(2r+1) = 0

$r_{1} = 1 , r_{2} = \frac{-1}{2}$

∴ there are two distinct roots

so the general equation is as follows:

y(x) = $C_{1} e^{r_{1}x } + C_{2} e^{r_{2}x }$

y(x) = $C_{1} e^{x} + C_{2} e^{\frac{-x}{2} }$