The solution of x d y d x +y=e^x is - Vidyadip

MCQ

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

✓
$y=\frac{e^x}{x}+\frac{c}{x}$
B
$y=x e^x+c x$
C
$y=x e^x+c$
D
$x=\frac{e^y}{y}+\frac{c}{y}$

✓

Answer

Correct option: A.

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

$x \frac{d y}{d x}+y=e^x$
$\frac{d y}{d x}+\frac{y}{x}=\frac{e^x}{x}$
It is a linear differential equation with
$\text { I.F. }=e^{\int \frac{1}{x} d x}=e^{\log x}=x$
Now, solution is $y \cdot x=\int \frac{e^x}{x} \cdot x d x+c$
$\Rightarrow y x=e^x+c$
$\Rightarrow y=\frac{e^x}{x}+\frac{c}{x}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "M.C.Q (1 Marks)" from Differential Equations→Differential Equations — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

If $x$ takes negative permissible value, then $\sin−1\text{x} $ is equal to:

Let $A$ be a $3 \times 3$ matrix with $\operatorname{det}( A )=4$. Let $R _{ i }$ denote the $i ^{\text {th }}$ row of $A$. If a matrix $B$ is obtained by performing the operation $R _{2} \rightarrow 2 R _{2}+5 R _{3}$ on $2 A ,$ then $\operatorname{det}( B )$ is equal to ...... .

If the co-ordinates of the points $A,B,C$ be $( - 1,\,3,\,2),\,\,(2,\,3,\,5)$ and $(3, 5,-2) $ respectively, then $\angle A = $ ..…… $^o$

If $x$ co-ordinates of a point $P$ of line joining the points $Q(2,\,2,\,1)$ and $R\,(5,\,2, - 2)$ is $4$, then the $z$ - coordinates of $P$ is

If $y = \frac{{5x}}{{\sqrt[3]{{{{(1 - x)}^2}}}}} + {\cos ^2}(2x + 1)$, then $\frac{{dy}}{{dx}} = $

The equation of the normal to the curve $y = x(2 - x)$ at the point $(2, 0)$ is:

$\int {\frac{{x\,\,dx}}{{{x^2} + 4x + 5}} = } $

If three points $A, B $ and $ C$ have position vectors $(1,\,x,\,3),\,\,(3,\,4,\,7)$ and $(y,-2-5)$ respectively and if they are collinear, then $(x,\,y) = $

If $2x + 5y - 6z + 3 = 0$ be the equation of the plane, then the equation of any plane parallel to the given plane is:

$\int\text{e}^\text{x}(\frac{1-\text{x}}{1+\text{x}^2})^2\text{dx}$ is equal to: