# Ordinary Differential Equations/Homogenous 4

1)

$3y''+18y'-81y=0$ Step 1: Get the equation in the form $C_{1}y^{(n)}+C_{2}y^{(n-1)}+...+C_{n+1}y=0$ $y''+6y'-27y=0$ Step 2: Find the roots of the equation $C_{1}r^{n}+C_{2}r^{n-1}+...+C_{n+1}$ $r^{2}+6r-27=0$ $r=-9,3$ Step 3: Your result is $y=c_{1}e^{r_{1}x}+c_{2}e^{r_{2}x}+...+c_{n}e^{r_{n}x}$ $y=c_{1}e^{-9x}+c_{2}e^{3x}$ 2)

$y''+6y'+13y=0$ Step 1: Get the equation in the form $C_{1}y^{(n)}+C_{2}y^{(n-1)}+...+C_{n+1}y=0$ $y''+6y'+13y=0$ Step 2: Find the roots of the equation $C_{1}r^{n}+C_{2}r^{n-1}+...+C_{n+1}$ $r^{2}+6r+13=0$ $r=-3\pm 2i$ Step 3: Your result is $y=c_{1}e^{r_{1}x}+c_{2}e^{r_{2}x}+...+c_{n}e^{r_{n}x}$ $y=e^{-3x}(c_{1}cos(2x)+c_{2}sin(2x))$ 3)$y''+10y'+25y=0$ Step 1: Get the equation in the form $C_{1}y^{(n)}+C_{2}y^{(n-1)}+...+C_{n+1}y=0$ $y''+10y'+25y=0$ Step 2: Find the roots of the equation $C_{1}r^{n}+C_{2}r^{n-1}+...+C_{n+1}$ $r^{2}+10r+25=0$ $r=-5,-5$ Step 3: Your result is $y=c_{1}e^{r_{1}x}+c_{2}e^{r_{2}x}+...+c_{n}e^{r_{n}x}$ $y=c_{1}e^{-5x}+c_{2}xe^{-5x}$ 4)

$y''''+24y'''+218y''+838y'+1369y=0$ Step 1: Get the equation in the form $C_{1}y^{(n)}+C_{2}y^{(n-1)}+...+C_{n+1}y=0$ $y''''+24y'''+218y''+838y'+1369y=0$ Step 2: Find the roots of the equation $C_{1}r^{n}+C_{2}r^{n-1}+...+C_{n+1}$ $r^{4}+24r^{3}+218r^{2}+838r+1369=0$ $r=-6-i,-6+i,-6-i,-6+i$ Step 3: Your result is $y=c_{1}e^{r_{1}x}+c_{2}e^{r_{2}x}+...+c_{n}e^{r_{n}x}$ $y=e^{-6x}(c_{1}cos(x)+c_{2}sin(x)+c_{3}xcos(x)+c_{4}xsin(x))$ 5)

$y'''-2y''-15y'+36y=0$ Step 1: Get the equation in the form $C_{1}y^{(n)}+C_{2}y^{(n-1)}+...+C_{n+1}y=0$ $y'''-2y''-15y'+36y=0$ Step 2: Find the roots of the equation $C_{1}r^{n}+C_{2}r^{n-1}+...+C_{n+1}$ $r^{3}-2r^{2}-15r+36=0$ $r=-4,3,3$ Step 3: Your result is $y=c_{1}e^{r_{1}x}+c_{2}e^{r_{2}x}+...+c_{n}e^{r_{n}x}$ $y=c_{1}e^{-4x}+c_{2}e^{3x}+c_{3}xe^{3x}$ 6)

$y'''+5y''-4y'-20y=0$ Step 1: Get the equation in the form $C_{1}y^{(n)}+C_{2}y^{(n-1)}+...+C_{n+1}y=0$ $y'''+5y''-4y'-20y=0$ Step 2: Find the roots of the equation $C_{1}r^{n}+C_{2}r^{n-1}+...+C_{n+1}$ $r^{3}+5r^{2}-4r-20=0$ $r=2,-2,-5$ Step 3: Your result is $y=c_{1}e^{r_{1}x}+c_{2}e^{r_{2}x}+...+c_{n}e^{r_{n}x}$ $y=c_{1}e^{2x}+c_{2}e^{-2x}+c_{3}e^{-5x}$ 7)

$y'''+4y''+y'-26y=0$ Step 1: Get the equation in the form $C_{1}y^{(n)}+C_{2}y^{(n-1)}+...+C_{n+1}y=0$ $y'''+4y''+y'-26y=0$ Step 2: Find the roots of the equation $C_{1}r^{n}+C_{2}r^{n-1}+...+C_{n+1}$ $r^{3}+4r^{2}+r-26=0$ $r=-3+2i,-3-2i,2$ Step 3: Your result is $y=c_{1}e^{r_{1}x}+c_{2}e^{r_{2}x}+...+c_{n}e^{r_{n}x}$ $y=c_{1}e^{2x}+e^{-3x}(c_{2}cos(2x)+c_{3}sin(2x))$ 