Question

The general solution of y000 −4 y00 + y0 + 6 y = 0 is: (Hint: r = 2 is a root of the characteristic equation)

Answer 1

Solution:

$\small {y}'''-4{y}''+{y}'+6y=0$

The characteristic equation is

$\small r^{^{3}}-4r^{2}+r+6=0$

Using synthetic division we have,

$\small 2$	$\small 1$	$\small -4$	$\small 1$	$\small 6$
		$\small 2$	$\small -4$	$\small -6$
	$\small 1$	- 2	$\small -3$	$\small 0$

$\small \therefore \left ( r-2 \right )\left ( r^{2}-2r-3 \right )=0$

$\small \therefore \left ( r-2 \right )\left ( r^{2}-3r+r-3 \right )=0$

$\small \therefore \left ( r-2 \right )\left [ r\left ( r-3 \right )+1\left ( r-3 \right ) \right ]=0$

$\small \therefore \left ( r-2 \right )\left ( r-3 \right )\left ( r+1 \right )=0$

$\small \therefore r=-1,2,3$

$\small \therefore$ The general solution of the given differential equation is

$\small y=c_{1}e^{-x}+c_{2}e^{2x}+c_{3}e^{3x}$