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 $\left(3.9\right)$, and Example .

$\mathit{y}\mathbf{+}\mathit{y}\mathbf{/}\sqrt{{\mathbf{x}}^{\mathbf{2}}\mathbf{+}\mathbf{1}}\mathbf{=}\mathbf{1}\mathbf{/}\left(x+\sqrt{x^2+1}\right)$

Short answer

Answer:

The general solution of the differential equation is $y-\frac{x+C}{x+\sqrt{x^{}+1}}$.

Step-by-step solution

Given information

The given differential equation is$y+y/\sqrt{x^2+1}=1/\left(x+\sqrt{x^2+1}\right)$.

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 $y+Py=Q$. From equation 3.4,

$$\begin{gathered} I=\int \frac{dx}{\sqrt{x^2+1}} \\ =In\left(x+\sqrt{x^2+1}\right)e^I \\ =x+\sqrt{x^2+1} \end{gathered}$$.

Find a solution for this differential equation.

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

Therefore, the general solution of the differential equation is $y=\frac{x+C}{x+\sqrt{x^2+1}}$.