Question

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

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

Short answer

The solution of the given Cauchy Euler differential equation

$2x^3{y}^{\text{'}\text{'}\text{'}}+19x^2{y}^{\text{'}\text{'}}+39x{y}^{\text{'}}+9y=0$as$y=c_1{x}^{-3}+c_2{x}^{-3}\ln x+c_3{x}^{-\frac{1}{2}}$

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.

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

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

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

Which is $y^{\text{'}\text{'}\text{'}}+\frac{19}{2}{x}^{-1}{y}^{\text{'}\text{'}}+\frac{39}{2}{x}^{-2}{y}^{\text{'}}+\frac{9}{2}{x}^{-3}y=0$

$y^{\text{'}\text{'}\text{'}}+p\left(x\right)y^{\text{'}\text{'}}+q\left(x\right)y^{\text{'}}+f\left(x\right)y=0,$

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

$y^{\text{'}}=m{x}^{(m-1)}$

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

Now substitute $y^{\text{'}},y^{\text{'}\text{'}}\text{and}{y}^{\text{'}\text{'}\text{'}}$

$2x^3\left(m(m-1)(m-2)x^{(m-3)}\right)+19x^2\left(m(m-1)x^{(m-2)}\right)+39x\left(m{x}^{(m-1)}\right)+9x^m=0$

$2m(m-1)(m-2)x^3·x^{(m-3)}+19m(m-1)x^2·x^{(m-2)}+39mx·x^{(m-1)}+9x^m=0$

$\left(2m^3+13m^2+24m+9\right)x^m=0$

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$m=-3$

role="math" localid="1667973313699" $$\begin{gathered} (m+3)\left(2m^2+7m+3\right)=0 \\ (m+3)(m+3)(2m+1)=0 \end{gathered}$$

$$\begin{gathered} m_{1,2}=-3 \\ {m}_3=-\frac{1}{2} \end{gathered}$$

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