The Solution of x dx + y = e x is

Given that x dx + y = e x dx + x = x which is a linear differential equation. = e x dx = e ( x) = x The general Solution is y x = x x ) dx y x = dx y x = e x + k y = x + x

Differential Equations — Class 12 Maths Solution

exemplar objective MCQ NCERT EXEMP.Q.58,Page.198

Question

The Solution of $x\frac{{dy}}{{dx}} + y = {e^x}$ is

(a) $y = \frac{{{e^x}}}{x} + \frac{k}{x}$ ✓ Correct
(b) $y = x{{\rm{e}}^x} + {\rm{Cx}}$
(c) $y = x{{\rm{e}}^x} + k$
(d) $x = \frac{{{{\rm{e}}^y}}}{y} + \frac{k}{y}$

Step-by-step Solution

Given that $x\frac{{dy}}{{dx}} + y = {e^x}$

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

which is a linear differential equation.

$\therefore {\rm{IF}} = {e^{\int {\frac{1}{x}} dx}} = {e^{(\log x)}} = x$

The general Solution is $y \cdot x = \int {\left( {\frac{{{d^x}}}{x} \cdot x} \right)} dx$

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

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

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

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