veksetz

2021-12-31

Identify the characteristic equation, solve for the characteristic roots, and solve the 2nd order differential equations.

$(18{D}^{3}-33{D}^{2}+20D-4)y=0$

Robert Pina

Beginner2022-01-01Added 42 answers

Finding characteristic equation by putting r at the place of D then solve r to find characteristic root

$(18{D}^{3}-33{D}^{2}+20D-4)y=0$

Characteristic equation

$18{r}^{3}-33{r}^{2}+20r-4=0$

Now solving the above characteristic equation to find characteristic root.

$18{r}^{3}-33{r}^{2}+20r-4=0$

put$r=\frac{1}{2}$

$18\times \frac{1}{8}-33\times \frac{1}{4}+20\times \frac{1}{2}-4$

$\frac{9}{4}-\frac{33}{4}+10-4$

$-6+6=0$

So$x=\frac{1}{2}$ is a characteristic root

Now$18{r}^{3}-33{r}^{2}+20r-4=0$

$18{r}^{2}(r-\frac{1}{2})-24r(r-\frac{1}{2})+8(r-\frac{1}{2})=0$

$(r-\frac{1}{2})[18{r}^{2}-24r+8]=0$

Now$18{r}^{2}-24r+8=0$

$9{r}^{2}-12r+4=0$

$9{r}^{2}-(6+6)r+4=0$

$9{r}^{2}-6r-6r+4=0$

$3r(3r-2)-2(3r-2)=0$

$(3r-2)(3r-2)=0$

$r=\frac{2}{3}$ (repeated)

Now characteristic roots are

$r=\frac{1}{2}\text{}{\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}\text{}r=\frac{2}{3}$ (repeated)

So solution for defferential equation will be

$y\left(x\right)={c}_{1}{e}^{\frac{1}{2}x}+({c}_{2}+{c}_{3}x){e}^{\frac{2}{3}x}$

here$c}_{1},{c}_{2},{c}_{3$ are arbitrary constant

Characteristic equation

Now solving the above characteristic equation to find characteristic root.

put

So

Now

Now

Now characteristic roots are

So solution for defferential equation will be

here

Donald Cheek

Beginner2022-01-02Added 41 answers

Therefore, $18{m}^{3}-33{m}^{2}+20m-4=0$

Where${m}^{3}={D}^{3}y,{m}^{2}=Dy,m=Dy$ , as y

$\Rightarrow (2m-1){(3m-2)}^{2}=0$

$\Rightarrow 2n-1=0\text{}on\text{}{(3m-2)}^{2}=0$

Hence the roots are$\frac{1}{2},\frac{2}{3},\frac{2}{3}$

So, the general solution of the given equation is

$y={c}_{1}{e}^{\frac{x}{2}}+({c}_{2}+{c}_{3}x){e}^{2\frac{x}{3}}$

Where

Hence the roots are

So, the general solution of the given equation is

Vasquez

Expert2022-01-09Added 669 answers

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My steps:

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