The solution of the differential equation x l | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) Given differential equation is $x \log x \frac{dy}{dx} + y = 2 \log x$.Dividing both sides by $x \log x$,we get:$\frac{dy}{dx} + \frac{1}{x \log x

Vedclass · Accepted Answer

$y \log x = (\log x)^2 + c$

Vedclass · Answer

(C) Given differential equation is $x \log x \frac{dy}{dx} + y = 2 \log x$.Dividing both sides by $x \log x$,we get:$\frac{dy}{dx} + \frac{1}{x \log x} y = \frac{2}{x}$.This is a linear differential equation of the form $\frac{dy}{dx} + Py = Q$,where $P = \frac{1}{x \log x}$ and $Q = \frac{2}{x}$.Integrating factor $(I.F.) = e^{\int P dx} = e^{\int \frac{1}{x \log x} dx}$.Let $\log x = t$,then $\frac{1}{x} dx = dt$.So,$I.F. = e^{\int \frac{1}{t} dt} = e^{\log t} = t = \log x$.The general solution is $y \cdot (I.F.) = \int Q \cdot (I.F.) dx + c$.$y \log x = \int \frac{2}{x} \cdot \log x dx$.Let $\log x = u$,then $\frac{1}{x} dx = du$.$y \log x = \int 2u du = u^2 + c$.Substituting $u = \log x$,we get $y \log x = (\log x)^2 + c$.

The solution of the differential equation $x \log x \frac{dy}{dx} + y = 2 \log x$ is

Similar Questions

The equation of the curve passing through $(1,2)$ and whose tangent at any point $(x, y)$ makes an angle $\tan ^{-1}(2 x+3 y)$ with the $X$-axis is .........

The function $y=f(x)$ is the solution of the differential equation $\frac{dy}{dx}+\frac{xy}{x^2-1}=\frac{x^4+2x}{\sqrt{1-x^2}}$ in $(-1,1)$ satisfying $f(0)=0$. Then $\int_{-\frac{\sqrt{3}}{2}}^{\frac{\sqrt{3}}{2}} f(x) dx$ is

Let $y=y(x)$ be the solution of the differential equation $x^{4}dy + (4x^{3}y + 2\sin x)dx = 0$, $x>0$, $y(\frac{\pi}{2})=0$. Then $\pi^{4}y(\frac{\pi}{3})$ is equal to:

The solution of the differential equation $(x + 2y^3) \frac{dy}{dx} - y = 0$ is

If $y = (A + Bx) e^{mx} + (m - 1)^{-2} e^x$,then $\frac{d^2y}{dx^2} - 2m \frac{dy}{dx} + m^2y$ is equal to:

Vedclass Test Series

Exam Paper Generator

Online Exam Module