The integrating factor of the differential equation (x ) dy dx +y… - Vidyadip

MCQ

The integrating factor of the differential equation $(\text{x}\log\text{x})\frac{\text{dy}}{\text{dx}}+\text{y}=2\ \log\text{x}$ is given by:

A
$\log(\log\text{x})$
B
$\text{e}^{\text{x}}$
✓
$\log\text{x}$
D
$\text{x}$

✓

Answer

Correct option: C.

$\log\text{x}$

We have,
$(\text{x}\log\text{x})\frac{\text{dy}}{\text{dx}}+\text{y}=2\ \log\text{x}$
Dividing both sides by,
$\frac{\text{dy}}{\text{dx}}+\frac{\text{y}}{\text{x}+\log\text{x}}=\frac{2}{\text{x}}$
$\frac{\text{dy}}{\text{dx}}+\Big(\frac{\text{1}}{\text{x}+\log\text{x}}\Big)\text{y}=\frac{2}{\text{x}}$
Comparing with $\frac{\text{dy}}{\text{dx}}+\text{Py}=\text{Q}$
$\text{P}=\frac{1}{\text{x}\log\text{x}}$
$\text{Q}=\frac{2}{\text{x}}$
Now,
$\text{I.F}=\text{e}^{\int\text{P}\text{dx}}$
$\text{I.F}=\text{e}^{\int\frac{1}{\text{x}\log\text{x}}}\text{dx}$
$=\text{e}^{\log(\log\text{x})}$
$=\log\text{x}$

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 the system of equations

$x+y+a z=b$

$2 x+5 y+2 z=6$

$x+2 y+3 z=3$

has infinitely many solutions, then $2 a+3 b$ is equal to $...........$.

$\int_{\,2}^{\,3} {\frac{{dx}}{{{x^2} - x}} = } $

A card is picked at random from a pack of 52 playing cards. Given that the picked card is a queen, the probability of this card to be a card of spade is

$\lim _{n \rightarrow \infty} \frac{3}{n}\left\{4+\left(2+\frac{1}{n}\right)^2+\left(2+\frac{2}{n}\right)^2+\ldots+\left(3-\frac{1}{n}\right)^2\right\}$ is equal to

The value of p and q for which the function $f ( x )=\left\{\begin{array}{ll}\frac{\sin (p+1) x+\sin x}{x} & , x<0 \\ q & , x=0 \\ \frac{\sqrt{x+b x^2}-\sqrt{x}}{x^{\frac{3}{2}}} & , x>0\end{array}\right.$ is continuous for all $x \in R$, are

If $\text{S}=\begin{bmatrix}\text{a} & \text{b} \\ \text{c} & \text{d} \end{bmatrix},$ then $\text{adj A}$ is :

The number of binary operation that can be defined on a set of 2 elements is:

Two events $A$ and $B$ will be independent, if

The point of intersection of the lines $\frac{{x + 1}}{3} = \frac{{y + 3}}{5} = \frac{{z + 5}}{7}$ and $\frac{{x - 2}}{1} = \frac{{y - 4}}{3} = \frac{{z - 6}}{5}$is

If $f(x) = \left\{ \begin{array}{l}\,\,\,\,\,\,\,\,\,\frac{{1 - \cos 4x}}{{{x^2}}},\;\;{\rm{when}}\,x < 0\\\,\,\,\,\,\,\,\,\,\,\,a,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\rm{when}}\,\,x = 0\\\frac{{\sqrt x }}{{\sqrt {(16 + \sqrt x )} - 4}},\,\,{\rm{when}}\,\, x > 0\end{array} \right.$, is continuous at $x = 0$, then the value of $'a'$ will be