Question

Use the methods of this section to solve the following differential equations. Compare computer solutions and reconcile differences.

$\mathit{y}\mathbf{\text{'}}\mathbf{=}\frac{\mathbf{y}}{\mathbf{x}}\mathbf{-}\mathit{t}\mathit{a}\mathit{n}\frac{\mathbf{y}}{\mathbf{x}}$

Short answer

Answer

The general solution of the differential equation is $y=x{\sin }^{-1}\frac{C_1}{x}$

Step-by-step solution

Given information

The given differential equation is$y\text{'}=\frac{y}{x}-\tan \frac{y}{x}$.

Definition of differential equation

A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself to its derivatives of various orders.

Solve the differential equation

Since $y\text{'}=f\frac{y}{x}$, therefore we can solve this differential equation as a homogeneous equation. Assume $v=\frac{y}{x}$. Then, $y\text{'}=xv\text{'}+v$. Therefore,

$$\begin{gathered} xv\text{'}+v=v-\tan v \\ v\text{'}=-\frac{\tan v}{x} \end{gathered}$$

, which has become a separable equation and its solution of is,

$$\begin{gathered} \frac{dv}{\tan v}=-\frac{dx}{x} \\ In\left(Sinv\right)=-Inx+C \\ \sin v=\frac{C_1}{x} \end{gathered}$$

Substitute $v=\frac{y}{x}$. Therefore, the solution becomes,

$y=x{\sin }^{-1}\frac{C_1}{x}$