The solution of x d y d x + y = e x is:

Question

The solution of $x \frac{d y}{d x} + y = e^{x}$ is:

Smart Achievers · Accepted Answer

(1)

Ans. $y = \frac{e^{x}}{x} + \frac{k}{x}$

Explanation:

Given that,

$x \frac{d y}{d x} + y = e^{x}$

$\Rightarrow$ $\frac{d y}{d x} + \frac{y}{x} = \frac{e^{x}}{x}$

which is a linear differential equation.

$∴$ $IF = e^{\int \frac{1}{x} d x} = e^{\log x} = x$

The general solution is

$\Rightarrow$ $y \cdot x = \int (\frac{e^{x}}{x} \cdot x) d x$

$\Rightarrow$ $y \cdot x = \int e^{x} d x$

$\Rightarrow$ $y \cdot x = e^{x} + k$

$\Rightarrow$ $y = \frac{e^{x}}{x} + \frac{k}{x}$

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