Question

Using(3.9) , 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()3.9 , and Example 1 .

$\mathbf{\text{dx}}\mathbf{+}\left(\text{x}-{\text{e}}^{\text{y}}\right)\mathbf{\text{dy}}\mathbf{=}\mathbf{\text{0}}$

Short answer

The general solution of the differential equations is$x=\frac{e^y}{2}+C{e}^{-y}$

Step-by-step solution

Given Information.

The given differential equations is$dx+\left(x-e^y\right)dy=0$

Meaning of the first-order differential equation.

A first-order differential equation is defined by two variablesxand y , and its function f(x,y) is defined on an XY-plane region.

Find the general solution.

Write this differential equation to make it in the form$y^{\text{'}}+Py=Q$, that is

$x^{\text{'}}+x=e^y$

From eq. 3.4 ,

$$\begin{gathered} I=\int 1dy \\ =y{e}^I \\ =e^y \end{gathered}$$

Find a solution for this differential equation

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

Therefore, the general solution of the differential equations is$x=\frac{e^y}{2}+C{e}^{-y}$.