The solution of x (dy/dx) + y = eˣ is

The correct option is y = eˣ/x + k/x.

To find the solution, let us analyze the given first-order differential equation step-by-step:
$$x\frac{dy}{dx}+y=e^x$$

Step 1: Rewrite in standard linear form
A first-order linear differential equation has the standard form:
$$\frac{dy}{dx}+P\left(x\right)y=Q\left(x\right)$$
To transform our given equation into this form, we divide both sides of the equation by $x$:
$$\frac{dy}{dx}+\frac{1}{x}y=\frac{e^x}{x}$$
By comparing, we can identify:
$$P\left(x\right)=\frac{1}{x}$$
and
$$Q\left(x\right)=\frac{e^x}{x}$$

Step 2: Find the Integrating Factor (I.F.)
The integrating factor is given by the formula:
$$\text{I.F.}=e^{\int P\left(x\right)dx}$$
Substituting $P\left(x\right)=\frac{1}{x}$:
$$\text{I.F.}=e^{\int \frac{1}{x}dx}=e^{\ln \left|x\right|}=x$$

Step 3: Solve the differential equation
The general solution of a linear differential equation is given by:
$$y\cdot \left(\text{I.F.}\right)=\int Q\left(x\right)\cdot \left(\text{I.F.}\right)dx+k$$
where $k$ is the constant of integration. Substituting our functions:
$$y\cdot x=\int \left(\frac{e^x}{x}\cdot x\right)dx+k$$
Simplifying inside the integral:
$$y\cdot x=\int e^{x}dx+k$$
Integrating $e^x$:
$$y\cdot x=e^x+k$$

Step 4: Solve for y
Divide both sides of the equation by $x$ to isolate $y$:
$$y=\frac{e^x}{x}+\frac{k}{x}$$
Thus, the final general solution is $y=e^x/x+k/x$.