Question

Using $\left(3.9\right)$, find the general solution of each of the following differential equations. Compare a computer solution and, if necessary, reconcile it with yours. Hint: See comments just after $\mathbf{(}\mathbf{3}\mathbf{.}\mathbf{9}\mathbf{)}$, and Example 1.

$\left(xlnx\right)\mathit{y}\mathbf{+}\mathit{y}\mathbf{=}\mathit{I}\mathit{n}\mathit{x}$

Short answer

Answer:

The general solution of the differential equation is $y=\frac{Inx}{2}+\frac{C}{Inx}$.

Step-by-step solution

Given information

The given differential equation is$\left(x\ln x\right)y+y=Inx$.

Meaning of the first-order differential equation

A first-order differential equation is defined by two variables, x and y, and its function $\mathit{f}\left(x,y\right)$is defined on an XY-plane region.

Find the general solution

Write this differential equation to make it in the form $y+Py=Q$; that is

$y\text{'}+\frac{y}{x\ln x}=\frac{1}{x}$.

From equation 3.4,

$$\begin{gathered} I=\int \frac{dx}{x\ln x} \\ =In\left(Inx\right)e^I \\ =E^{In\left(Inx\right)} \\ =Inx \end{gathered}$$.

Find a solution for this differential equation.

$$\begin{gathered} y{e}^I=\int \left(\frac{1}{x}\right)Inxdx \\ =\frac{I{n}^{2}x}{2}+C \\ y=\frac{Inx}{2}+\frac{C}{Inx} \end{gathered}$$

Therefore, the general solution of the differential equation is$y=\frac{Inx}{2}+\frac{C}{Inx}$.