Question

In problems 63 and 64 Use a CAS to complete all derivatives and to carry out the simplification needed to verify that the indicated function is a particular solution of the given differential equation.

${\mathit{x}}^{\mathbf{3}}\mathit{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{2}{\mathit{x}}^{\mathbf{2}}\mathit{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{20}\mathit{x}\mathit{y}\mathbf{\text{'}}\mathbf{-}\mathbf{780}\mathit{y}\mathbf{=}\mathbf{0}$

$\mathit{y}\mathbf{=}\mathbf{20}\frac{\mathbf{\text{cos}}\mathbf{}\left(5\text{ln}x\right)}{\mathbf{x}}\mathbf{-}\mathbf{3}\frac{\mathbf{\text{sin}}\mathbf{}\left(5\text{ln}x\right)}{\mathbf{x}}$.

Short answer

The indicated function is a particular solution of the given differential equation.

Step-by-step solution

Definition of Ordinary Differential Equation

A system of ordinary differential equations is two or more equations involving the derivatives of two or more unknown functions of a single independent variable.

Compute derivatives

Given $y=20\frac{\cos (5\ln x)}{x}-3\frac{\sin (5\ln x)}{x}$

Let compute the first third derivatives of the function

$$\begin{gathered} y\text{'}=-\frac{97\sin \left(5\ln \left(x\right)\right)+35\text{cos}\left(5\ln \left(x\right)\right)}{x^2} \\ y\text{'}\text{'}={\left(-\frac{97\sin \left(5\ln \left(x\right)\right)+35\text{cos}\left(5\ln \left(x\right)\right)}{x^2}\right)}^{\text{'}} \\ y\text{'}\text{'}=\frac{369\sin \left(5\ln \left(x\right)\right)-415\text{cos}\left(5\ln \left(x\right)\right)}{x^3} \\ y\text{'}\text{'}\text{'}={\left(\frac{369\sin \left(5\ln \left(x\right)\right)-415\text{cos}\left(5\ln \left(x\right)\right)}{x^3}\right)}^{\text{'}} \\ y\text{'}\text{'}\text{'}=\frac{968\sin \left(5\ln \left(x\right)\right)+3090\text{cos}\left(5\ln \left(x\right)\right)}{x^4} \end{gathered}$$

Substitute the derivatives in given equation

Now substitute the derivatives into the given equation.

If we get equality, to be precise, if the right side of the equation is equal to zero, then we conclude that the function is indeed a particular solution to the DE.

$x^{3}y\text{'}\text{'}\text{'}+2x^{2}y\text{'}\text{'}+20xy\text{'}-780y=0$

$$\begin{gathered} x^3\frac{968\sin \left(5\ln \left(x\right)\right)+3090\text{cos}\left(5\ln \left(x\right)\right)}{x^4}+2x^2\frac{369\sin \left(5\ln \left(x\right)\right)-415\text{cos}\left(5\ln \left(x\right)\right)}{x^3} \\ +20x\left(-\frac{97\sin (5\ln (x\left)\right)+35cos(5ln(x\left)\right)}{x^2}\right)-780\left(20\frac{cos(5lnx)}{x}\right)-3\frac{sin(5lnx)}{x})=0 \end{gathered}$$

$$\begin{gathered} \frac{968\sin (5\ln \left(x\right))+2·369\text{sin}(5\ln \left(x\right)-20·97\sin \left(5\ln \left(x\right)\right)+3·78\text{sin}\left(5\ln \left(x\right)\right)}{x} \\ +\frac{3090\cos (5\ln \left(x\right))-2·415\text{cos}(5\ln \left(x\right)-20·35\cos \left(5\ln \left(x\right)\right)-78·20\text{cos}\left(5\ln \left(x\right)\right)}{x}=0 \end{gathered}$$

$$\begin{gathered} \frac{0}{x}+\frac{0}{x}=0 \\ 0=0 \end{gathered}$$

Therefore, the indicated function is a particular solution of the give differential equation.