Ordinary Differential Equations/Homogenous 4

1)

$$3y''+18y'-81y=0$$

Step 1: Get the equation in the form $$C_1y^{(n)}+C_2y^{(n-1)}+...+C_{n+1}y=0$$

$$y''+6y'-27y=0$$

Step 2: Find the roots of the equation $$C_1r^n+C_2r^{n-1}+...+C_{n+1}$$

$$r^2+6r-27=0$$

$$r=-9,3$$

Step 3: Your result is $$y=c_1e^{r_1x}+c_2e^{r_2x}+...+c_ne^{r_nx}$$

$$y=c_1e^{-9x}+c_2e^{3x}$$

2)

$$y''+6y'+13y=0$$

Step 1: Get the equation in the form $$C_1y^{(n)}+C_2y^{(n-1)}+...+C_{n+1}y=0$$

$$y''+6y'+13y=0$$

Step 2: Find the roots of the equation $$C_1r^n+C_2r^{n-1}+...+C_{n+1}$$

$$r^2+6r+13=0$$

$$r=-3 \pm 2i$$

Step 3: Your result is $$y=c_1e^{r_1x}+c_2e^{r_2x}+...+c_ne^{r_nx}$$

$$y=e^{-3x}(c_1cos(2x)+c_2sin(2x))$$

3)$$y''+10y'+25y=0$$

Step 1: Get the equation in the form $$C_1y^{(n)}+C_2y^{(n-1)}+...+C_{n+1}y=0$$

$$y''+10y'+25y=0$$

Step 2: Find the roots of the equation $$C_1r^n+C_2r^{n-1}+...+C_{n+1}$$

$$r^2+10r+25=0$$

$$r=-5,-5$$

Step 3: Your result is $$y=c_1e^{r_1x}+c_2e^{r_2x}+...+c_ne^{r_nx}$$

$$y=c_1e^{-5x}+c_2xe^{-5x}$$

4)

$$y'+24y+218y''+838y'+1369y=0$$

Step 1: Get the equation in the form $$C_1y^{(n)}+C_2y^{(n-1)}+...+C_{n+1}y=0$$

$$y'+24y+218y''+838y'+1369y=0$$

Step 2: Find the roots of the equation $$C_1r^n+C_2r^{n-1}+...+C_{n+1}$$

$$r^4+24r^3+218r^2+838r+1369=0$$

$$r=-6-i,-6+i,-6-i,-6+i$$

Step 3: Your result is $$y=c_1e^{r_1x}+c_2e^{r_2x}+...+c_ne^{r_nx}$$

$$y=e^{-6x}(c_1cos(x)+c_2sin(x)+c_3xcos(x)+c_4xsin(x))$$

5)

$$y'-2y-15y'+36y=0$$

Step 1: Get the equation in the form $$C_1y^{(n)}+C_2y^{(n-1)}+...+C_{n+1}y=0$$

$$y'-2y-15y'+36y=0$$

Step 2: Find the roots of the equation $$C_1r^n+C_2r^{n-1}+...+C_{n+1}$$

$$r^3-2r^2-15r+36=0$$

$$r=-4,3,3$$

Step 3: Your result is $$y=c_1e^{r_1x}+c_2e^{r_2x}+...+c_ne^{r_nx}$$

$$y=c_1e^{-4x}+c_2e^{3x}+c_3xe^{3x}$$

6)

$$y'+5y-4y'-20y=0$$

Step 1: Get the equation in the form $$C_1y^{(n)}+C_2y^{(n-1)}+...+C_{n+1}y=0$$

$$y'+5y-4y'-20y=0$$

Step 2: Find the roots of the equation $$C_1r^n+C_2r^{n-1}+...+C_{n+1}$$

$$r^3+5r^2-4r-20=0$$

$$r=2,-2,-5$$

Step 3: Your result is $$y=c_1e^{r_1x}+c_2e^{r_2x}+...+c_ne^{r_nx}$$

$$y=c_1e^{2x}+c_2e^{-2x}+c_3e^{-5x}$$

7)

$$y'+4y+y'-26y=0$$

Step 1: Get the equation in the form $$C_1y^{(n)}+C_2y^{(n-1)}+...+C_{n+1}y=0$$

$$y'+4y+y'-26y=0$$

Step 2: Find the roots of the equation $$C_1r^n+C_2r^{n-1}+...+C_{n+1}$$

$$r^3+4r^2+r-26=0$$

$$r=-3+2i, -3-2i,2$$

Step 3: Your result is $$y=c_1e^{r_1x}+c_2e^{r_2x}+...+c_ne^{r_nx}$$

$$y=c_1e^{2x}+e^{-3x}(c_2cos(2x)+c_3sin(2x))$$