The solution of the differential equation x f | vedclass

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

Question

(B) Given differential equation is $x \frac{dy}{dx} = y - x 	an \left(\frac{y}{x}ight)$.Dividing by $x$,we get $\frac{dy}{dx} = \frac{y}{x} - 	an

Vedclass · Accepted Answer

$y = x \sin^{-1}\left(\frac{k}{x}\right)$

Vedclass · Answer

(B) Given differential equation is $x \frac{dy}{dx} = y - x 	an \left(\frac{y}{x}ight)$.Dividing by $x$,we get $\frac{dy}{dx} = \frac{y}{x} - 	an \left(\frac{y}{x}ight)$.Let $y = vx$,then $\frac{dy}{dx} = v + x \frac{dv}{dx}$.Substituting these into the equation: $v + x \frac{dv}{dx} = v - 	an v$.This simplifies to $x \frac{dv}{dx} = -	an v$,or $\frac{dv}{	an v} = -\frac{dx}{x}$.Integrating both sides: $\int \cot v \, dv = -\int \frac{1}{x} \, dx$.This gives $\ln |\sin v| = -\ln |x| + \ln |k|$.Using logarithm properties,$\ln |\sin v| = \ln \left|\frac{k}{x}ight|$,so $\sin v = \frac{k}{x}$.Substituting $v = \frac{y}{x}$,we get $\sin \left(\frac{y}{x}ight) = \frac{k}{x}$,which implies $y = x \sin^{-1} \left(\frac{k}{x}ight)$.Thus,option $B$ is correct.

The solution of the differential equation $x \frac{dy}{dx} = y - x \tan \left(\frac{y}{x}\right)$ is (Here,$k$ is an arbitrary constant)

Similar Questions

The solution of the differential equation $x dy - y dx = \sqrt{x^2+y^2} dx$, given that $y=1$ when $x=\sqrt{3}$, is

If $x \sin \left(\frac{y}{x}\right) dy = \left[y \sin \left(\frac{y}{x}\right) - x\right] dx$, $x > 0$ and $y(1) = \frac{\pi}{2}$, then the value of $\cos \left(\frac{y}{x}\right)$ is

Find the particular solution satisfying the given condition: $\left[x \sin ^{2}\left(\frac{y}{x}\right)-y\right] d x+x d y=0$; $y=\frac{\pi}{4}$ when $x=1$.

Show that the differential equation $x dy - y dx = \sqrt{x^{2} + y^{2}} dx$ is a homogeneous equation and find its solution.

Show that the differential equation $y^{\prime} = \frac{x+y}{x}$ is a homogeneous equation and find its general solution.

Vedclass Test Series

Exam Paper Generator

Online Exam Module