1)

${\displaystyle 3y''+18y'-81y=0}$

Step 1: Get the equation in the form ${\displaystyle C_{1}y^{(n)}+C_{2}y^{(n-1)}+...+C_{n+1}y=0}$

${\displaystyle y''+6y'-27y=0}$

Step 2: Find the roots of the equation ${\displaystyle C_{1}r^{n}+C_{2}r^{n-1}+...+C_{n+1}}$

${\displaystyle r^{2}+6r-27=0}$

${\displaystyle r=-9,3}$

Step 3: Your result is ${\displaystyle y=c_{1}e^{r_{1}x}+c_{2}e^{r_{2}x}+...+c_{n}e^{r_{n}x}}$

${\displaystyle y=c_{1}e^{-9x}+c_{2}e^{3x}}$

2)

${\displaystyle y''+6y'+13y=0}$

Step 1: Get the equation in the form ${\displaystyle C_{1}y^{(n)}+C_{2}y^{(n-1)}+...+C_{n+1}y=0}$

${\displaystyle y''+6y'+13y=0}$

Step 2: Find the roots of the equation ${\displaystyle C_{1}r^{n}+C_{2}r^{n-1}+...+C_{n+1}}$

${\displaystyle r^{2}+6r+13=0}$

${\displaystyle r=-3\pm 2i}$

Step 3: Your result is ${\displaystyle y=c_{1}e^{r_{1}x}+c_{2}e^{r_{2}x}+...+c_{n}e^{r_{n}x}}$

${\displaystyle y=e^{-3x}(c_{1}cos(2x)+c_{2}sin(2x))}$

3)${\displaystyle y''+10y'+25y=0}$

Step 1: Get the equation in the form ${\displaystyle C_{1}y^{(n)}+C_{2}y^{(n-1)}+...+C_{n+1}y=0}$

${\displaystyle y''+10y'+25y=0}$

Step 2: Find the roots of the equation ${\displaystyle C_{1}r^{n}+C_{2}r^{n-1}+...+C_{n+1}}$

${\displaystyle r^{2}+10r+25=0}$

${\displaystyle r=-5,-5}$

Step 3: Your result is ${\displaystyle y=c_{1}e^{r_{1}x}+c_{2}e^{r_{2}x}+...+c_{n}e^{r_{n}x}}$

${\displaystyle y=c_{1}e^{-5x}+c_{2}xe^{-5x}}$

4)

${\displaystyle y''''+24y'''+218y''+838y'+1369y=0}$

Step 1: Get the equation in the form ${\displaystyle C_{1}y^{(n)}+C_{2}y^{(n-1)}+...+C_{n+1}y=0}$

${\displaystyle y''''+24y'''+218y''+838y'+1369y=0}$

Step 2: Find the roots of the equation ${\displaystyle C_{1}r^{n}+C_{2}r^{n-1}+...+C_{n+1}}$

${\displaystyle r^{4}+24r^{3}+218r^{2}+838r+1369=0}$

${\displaystyle r=-6-i,-6+i,-6-i,-6+i}$

Step 3: Your result is ${\displaystyle y=c_{1}e^{r_{1}x}+c_{2}e^{r_{2}x}+...+c_{n}e^{r_{n}x}}$

${\displaystyle y=e^{-6x}(c_{1}cos(x)+c_{2}sin(x)+c_{3}xcos(x)+c_{4}xsin(x))}$

5)

${\displaystyle y'''-2y''-15y'+36y=0}$

Step 1: Get the equation in the form ${\displaystyle C_{1}y^{(n)}+C_{2}y^{(n-1)}+...+C_{n+1}y=0}$

${\displaystyle y'''-2y''-15y'+36y=0}$

Step 2: Find the roots of the equation ${\displaystyle C_{1}r^{n}+C_{2}r^{n-1}+...+C_{n+1}}$

${\displaystyle r^{3}-2r^{2}-15r+36=0}$

${\displaystyle r=-4,3,3}$

Step 3: Your result is ${\displaystyle y=c_{1}e^{r_{1}x}+c_{2}e^{r_{2}x}+...+c_{n}e^{r_{n}x}}$

${\displaystyle y=c_{1}e^{-4x}+c_{2}e^{3x}+c_{3}xe^{3x}}$

6)

${\displaystyle y'''+5y''-4y'-20y=0}$

Step 1: Get the equation in the form ${\displaystyle C_{1}y^{(n)}+C_{2}y^{(n-1)}+...+C_{n+1}y=0}$

${\displaystyle y'''+5y''-4y'-20y=0}$

Step 2: Find the roots of the equation ${\displaystyle C_{1}r^{n}+C_{2}r^{n-1}+...+C_{n+1}}$

${\displaystyle r^{3}+5r^{2}-4r-20=0}$

${\displaystyle r=2,-2,-5}$

Step 3: Your result is ${\displaystyle y=c_{1}e^{r_{1}x}+c_{2}e^{r_{2}x}+...+c_{n}e^{r_{n}x}}$

${\displaystyle y=c_{1}e^{2x}+c_{2}e^{-2x}+c_{3}e^{-5x}}$

7)

${\displaystyle y'''+4y''+y'-26y=0}$

Step 1: Get the equation in the form ${\displaystyle C_{1}y^{(n)}+C_{2}y^{(n-1)}+...+C_{n+1}y=0}$

${\displaystyle y'''+4y''+y'-26y=0}$

Step 2: Find the roots of the equation ${\displaystyle C_{1}r^{n}+C_{2}r^{n-1}+...+C_{n+1}}$

${\displaystyle r^{3}+4r^{2}+r-26=0}$

${\displaystyle r=-3+2i,-3-2i,2}$

Step 3: Your result is ${\displaystyle y=c_{1}e^{r_{1}x}+c_{2}e^{r_{2}x}+...+c_{n}e^{r_{n}x}}$

${\displaystyle y=c_{1}e^{2x}+e^{-3x}(c_{2}cos(2x)+c_{3}sin(2x))}$