Ordinary Differential Equations/Homogenous 4

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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))