Question

2. Find the solution of each differential equation. (a) y/2y-8y = 0 (b) 25y/- 20y + 4y = 0 (c) y + 2y + 2y = 0 2. Find the solution of each differential equation . ( a ) y / 2y - 8y = 0 ( b ) 25y / - 20y + 4y = 0 ( c ) y + 2y + 2y = 0​

Answer 1

Each of these ODEs is linear and homogeneous with constant coefficients, so we only need to find the roots to their respective characteristic equations.

(a) The characteristic equation for

$y'' - 2y' - 8y = 0$

is

$r^2 - 2r - 8 = (r - 4) (r + 2) = 0$

which arises from the ansatz $y = e^{rx}$.

The characteristic roots are $r=4$ and $r=-2$. Then the general solution is

$\boxed{y = C_1 e^{4x} + C_2 e^{-2x}}$

where $C_1,C_2$ are arbitrary constants.

(b) The characteristic equation here is

$25r^2 - 20r + 4 = (5r - 2)^2 = 0$

with a root at $r=\frac25$ of multiplicity 2. Then the general solution is

$\boxed{y = C_1 e^{2/5\,x} + C_2 x e^{2/5\,x}}$

(c) The characteristic equation is

$r^2 + 2r + 2 = (r + 1)^2 + 1 = 0$

with roots at $r = -1 \pm i$, where $i=\sqrt{-1}$. Then the general solution is

$y = C_1 e^{(-1+i)x} + C_2 e^{(-1-i)x}$

Recall Euler's identity,

$e^{ix} = \cos(x) + i \sin(x)$

Then we can rewrite the solution as

$y = C_1 e^{-x} (\cos(x) + i \sin(x)) + C_2 e^{-x} (\cos(x) - i \sin(x))$

or even more simply as

$\boxed{y = C_1 e^{-x} \cos(x) + C_2 e^{-x} \sin(x)}$