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 .

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

Short answer

Answer:

The general solution of the differential equation is $y=\frac{e^x\left(1+e^x+{\displaystyle \frac{e^{2x}}{3}}\right)+C}{{\left(1+e^e\right)}^2}$.

Step-by-step solution

Given information

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

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

$y+\frac{2e^x}{1+e^x}y=e^x$.

From equation 3.4,

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

Find a solution for this differential equation.

$$\begin{gathered} y{e}^I=\int \left(e^x\right){\left(1+e^x\right)}^{2}dx \\ =e^x+e^{2x}+\frac{e^{3x}}{3}+Cy{\left(1+e^x\right)}^2 \\ =e^x\left(1+e^x+\frac{e^{2x}}{3}\right)+C \\ y=\frac{e^x\left(1+e^x+{\displaystyle \frac{e^{2x}}{3}}\right)+C}{{\left(1+e^x\right)}^2} \end{gathered}$$

Therefore, the general solution of the differential equation is $y=\frac{e^x\left(1+e^x+{\displaystyle \frac{e^{2x}}{3}}\right)+C}{{\left(1+e^x\right)}^2}$.