The solution of the equation xfrac{dy}{dx} = | vedclass

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

Question

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

Vedclass · Accepted Answer

$x\sin \left( \frac{y}{x} \right) = c$

Vedclass · Answer

(C) Given the differential equation: $x\frac{dy}{dx} = y - x	an \left( \frac{y}{x} ight)$.Dividing by $x$,we get: $\frac{dy}{dx} = \frac{y}{x} - 	an \left( \frac{y}{x} ight)$.This is a homogeneous differential equation. Let $y = vx$,then $\frac{dy}{dx} = v + x\frac{dv}{dx}$.Substituting these into the equation: $v + x\frac{dv}{dx} = v - 	an v$.Simplifying: $x\frac{dv}{dx} = -	an v$.Separating the variables: $\frac{dv}{	an v} = -\frac{dx}{x}$,which is $\cot v \, dv = -\frac{dx}{x}$.Integrating both sides: $\int \cot v \, dv = -\int \frac{dx}{x}$.$\ln |\sin v| = -\ln |x| + \ln |c|$.$\ln |\sin v| + \ln |x| = \ln |c|$.$\ln |x \sin v| = \ln |c|$.$x \sin v = c$.Substituting $v = \frac{y}{x}$ back: $x \sin \left( \frac{y}{x} ight) = c$.

The solution of the equation $x\frac{dy}{dx} = y - x\tan \left( \frac{y}{x} \right)$ is

Similar Questions

If $\cos \frac{y}{x} = A \log x + C$ is the general solution of $(x \sin \frac{y}{x}) dy = (y \sin \frac{y}{x} - x) dx$,then $A =$

The general solution of the differential equation $\frac{dy}{dx} = \frac{y + \sqrt{x^2 - y^2}}{x}$ is

The equation of the curve which passes through point $(1,0)$ and has a tangent with slope $1+\frac{y}{x}+\left(\frac{y}{x}\right)^{2}$ is

The general solution of the differential equation $(y^2-x^2) dx = xy dy$ $(x \neq 0)$ is

For the differential equation $(x^2 + y^2) dy = xy dx$, it is given that $y(1) = 1$ and $y(x_0) = e$. Find the value of $x_0$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module