Ordinary Differential Equations/Homogenous 4

1)

$3y^{″}+18y^{\prime }-81y=0$

Step 1: Get the equation in the form $C_1{y}^{(n)}+C_2{y}^{(n-1)}+...+C_{n+1}y=0$

$y^{″}+6y^{\prime }-27y=0$

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)

$y^{″}+6y^{\prime }+13y=0$

Step 1: Get the equation in the form $C_1{y}^{(n)}+C_2{y}^{(n-1)}+...+C_{n+1}y=0$

$y^{″}+6y^{\prime }+13y=0$

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)$y^{″}+10y^{\prime }+25y=0$

Step 1: Get the equation in the form $C_1{y}^{(n)}+C_2{y}^{(n-1)}+...+C_{n+1}y=0$

$y^{″}+10y^{\prime }+25y=0$

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 $y=c_1{e}^{r_{1}x}+c_2{e}^{r_{2}x}+...+c_n{e}^{r_{n}x}$

$y=c_1{e}^{-5x}+c_{2}x{e}^{-5x}$

4)

$y^{⁗}+24y^{‴}+218y^{″}+838y^{\prime }+1369y=0$

Step 1: Get the equation in the form $C_1{y}^{(n)}+C_2{y}^{(n-1)}+...+C_{n+1}y=0$

$y^{⁗}+24y^{‴}+218y^{″}+838y^{\prime }+1369y=0$

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)

$y^{‴}-2y^{″}-15y^{\prime }+36y=0$

Step 1: Get the equation in the form $C_1{y}^{(n)}+C_2{y}^{(n-1)}+...+C_{n+1}y=0$

$y^{‴}-2y^{″}-15y^{\prime }+36y=0$

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 $y=c_1{e}^{r_{1}x}+c_2{e}^{r_{2}x}+...+c_n{e}^{r_{n}x}$

$y=c_1{e}^{-4x}+c_2{e}^{3x}+c_{3}x{e}^{3x}$

6)

$y^{‴}+5y^{″}-4y^{\prime }-20y=0$

Step 1: Get the equation in the form $C_1{y}^{(n)}+C_2{y}^{(n-1)}+...+C_{n+1}y=0$

$y^{‴}+5y^{″}-4y^{\prime }-20y=0$

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)

$y^{‴}+4y^{″}+y^{\prime }-26y=0$

Step 1: Get the equation in the form $C_1{y}^{(n)}+C_2{y}^{(n-1)}+...+C_{n+1}y=0$

$y^{‴}+4y^{″}+y^{\prime }-26y=0$

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 $y=c_1{e}^{r_{1}x}+c_2{e}^{r_{2}x}+...+c_n{e}^{r_{n}x}$

$y=c_1{e}^{2x}+e^{-3x}(c_{2}cos(2x)+c_{3}sin(2x))$