Differential Equations/Homogenous 4

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1)

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

Step 1: Get the equation in the form $C 1 y (n) + C 2 y (n - 1) + ... + C n + 1 y = 0$

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

Step 2: Find the roots of the equation $C 1 r n + C 2 r n - 1 + ... + C n + 1$

$r 2 + 6 r - 27 = 0$

$r = - 9,3$

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

$y = c 1 e - 9 x + c 2 e 3 x$

2)

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

Step 1: Get the equation in the form $C 1 y (n) + C 2 y (n - 1) + ... + C n + 1 y = 0$

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

Step 2: Find the roots of the equation $C 1 r n + C 2 r n - 1 + ... + C n + 1$

$r 2 + 6 r + 13 = 0$

$r=-3 \pm 2i$

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

$y = e - 3 x (c 1 c o s (2 x) + c 2 s i n (2 x))$

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

Step 1: Get the equation in the form $C 1 y (n) + C 2 y (n - 1) + ... + C n + 1 y = 0$

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

Step 2: Find the roots of the equation $C 1 r n + C 2 r n - 1 + ... + C n + 1$

$r 2 + 10 r + 25 = 0$

$r = - 5, - 5$

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

$y = c 1 e - 5 x + c 2 x e - 5 x$

4)

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

Step 1: Get the equation in the form $C 1 y (n) + C 2 y (n - 1) + ... + C n + 1 y = 0$

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

Step 2: Find the roots of the equation $C 1 r n + C 2 r n - 1 + ... + C n + 1$

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

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

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

$y = e - 6 x (c 1 c o s (x) + c 2 s i n (x) + c 3 x c o s (x) + c 4 x s i n (x))$

5)

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

Step 1: Get the equation in the form $C 1 y (n) + C 2 y (n - 1) + ... + C n + 1 y = 0$

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

Step 2: Find the roots of the equation $C 1 r n + C 2 r n - 1 + ... + C n + 1$

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

$r = - 4,3,3$

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

$y = c 1 e - 4 x + c 2 e 3 x + c 3 x e 3 x$

6)

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

Step 1: Get the equation in the form $C 1 y (n) + C 2 y (n - 1) + ... + C n + 1 y = 0$

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

Step 2: Find the roots of the equation $C 1 r n + C 2 r n - 1 + ... + C n + 1$

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

$r = 2, - 2, - 5$

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

$y = c 1 e 2 x + c 2 e - 2 x + c 3 e - 5 x$

7)

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

Step 1: Get the equation in the form $C 1 y (n) + C 2 y (n - 1) + ... + C n + 1 y = 0$

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

Step 2: Find the roots of the equation $C 1 r n + C 2 r n - 1 + ... + C n + 1$

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

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

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

$y = c 1 e 2 x + e - 3 x (c 2 c o s (2 x) + c 3 s i n (2 x))$

Differential Equations/Homogenous 4

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