Question

use the procedures developed in this chapter to find the general solution of each differential equation.

$\mathbf{2}{\mathit{x}}^{\mathbf{3}}\mathit{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{19}{\mathit{x}}^{\mathbf{2}}\mathit{y}\mathbf{"}\mathbf{+}\mathbf{39}\mathit{x}\mathit{y}\mathbf{\text{'}}\mathbf{}\mathbf{+}\mathbf{9}\mathit{y}\mathbf{=}\mathbf{0}$

Short answer

The solution of the given Cauchy Euler differential equation

$2x^{3}y\text{'}\text{'}\text{'}+19x^{2}y"+39xy\text{'}+9y=0asy=c_1{x}^{-3}+c_2{x}^{-3}lnx+c_3{x}^{-\frac{1}{2}}$

Step-by-step solution

Define the Third order Cauchy Euler differential equation

The Cauchy-Euler differential equation is a type of linear ordinary differential equation with variable coefficients. In time-harmonic vibrations of a thin elastic rod, issues on yearly and solid discs, wave mechanics, and other domains of science and engineering, the second order Cauchy–Euler equations are applied.

${\mathit{a}}_{\mathbf{n}}{\mathit{x}}_{\mathbf{n}}{\mathit{y}}^{\left(n\right)}\mathbf{+}{\mathit{a}}_{\mathbf{n}\mathbf{-}\mathbf{1}}{\mathit{x}}^{\mathbf{n}\mathbf{-}\mathbf{1}}{\mathit{y}}^{(n-1)}\mathbf{+}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{+}{\mathit{a}}_{\mathbf{0}}\mathit{y}\mathbf{=}\mathbf{0}$

Step 2: Obtain the solution of the given Cauchy Euler differential equation

Here $2x^{3}y\text{'}\text{'}\text{'}+19x^{2}y"+39xy\text{'}+9y=0$

Which is $$\begin{gathered} y"\text{'}+\frac{19}{2}{x}^{-1}y"+\frac{39}{2}{x}^{-2}y\text{'}+\frac{9}{2}{x}^{-3}y=0 \\ y\text{'}\text{'}\text{'}+p\left(x\right)y"+q\left(x\right)y\text{'}+f\left(x\right)y=0 \end{gathered}$$

Now obtain the solution by assuming $y=x^m$

$$\begin{gathered} y\text{'}=m{x}^{\left(m-1\right)} \\ y"=m\left(m-1\right)x^{\left(m-2\right)} \\ y\text{'}\text{'}\text{'}=m\left(m-1\right)\left(m-2\right)x^{\left(m-3\right)} \end{gathered}$$

Now substitute $y\text{'},y\text{'}\text{'},andy\text{'}\text{'}\text{'}$

$$\begin{gathered} 2x^3\left(m\left(m-1\right)\right)\left(m-2\right)x^{\left(m-3\right)}+19x^2\left(m\left(m-1\right)x^{\left(m-2\right)}\right)+39x\left(m{x}^{\left(m-1\right)}\right)+9x^m=0 \\ 2m\left(m-1\right)\left(m-2\right)x^3.x^{\left(m-3\right)}+19m\left(m-1\right)x^2.x^{\left(m-2\right)}+39mx.x^{\left(m-1\right)}+9x^m=0 \\ \left(2m^3+13m^2+24m+9\right)x^m=0 \end{gathered}$$

Step 3: Obtain the solution when m=-1 or m =-2

As$x^m$cannot be 0.

$2m^3+13m^2+24m+9=0$

When, $m=-3$

$$\begin{gathered} \left(m+3\right)\left(2m^3+7m+3\right)=0 \\ \left(m+3\right)\left(m+3\right)\left(2m+1\right)=0 \\ {m}_{1,2}=-3 \\ {m}_3=-\frac{1}{2} \end{gathered}$$

The solution is$y=c_1{x}^{-3}+c_2{x}^{-3}\ln x+c_3{x}^{-\frac{1}{2}}$