The general solution of the differential equa | vedclass

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

Question

(B) Given differential equation: $\left(x \sin \frac{y}{x}\right) dy = \left(y \sin \frac{y}{x} - x\right) dx$ Rearranging the terms: $\frac{dy}{dx} =

Vedclass · Accepted Answer

$\cos\left(\frac{y}{x}\right) = \log |x| + c$

Vedclass · Answer

(B) Given differential equation: $\left(x \sin \frac{y}{x}ight) dy = \left(y \sin \frac{y}{x} - xight) dx$ Rearranging the terms: $\frac{dy}{dx} = \frac{y \sin \frac{y}{x} - x}{x \sin \frac{y}{x}} = \frac{y}{x} - \frac{1}{\sin(y/x)} = \frac{y}{x} - 	ext{cosec}\left(\frac{y}{x}ight)$ Let $y = vx$, then $\frac{dy}{dx} = v + x \frac{dv}{dx}$. Substituting these into the equation: $v + x \frac{dv}{dx} = v - 	ext{cosec}(v)$ $x \frac{dv}{dx} = -	ext{cosec}(v)$ Separating the variables: $\sin(v) dv = -\frac{1}{x} dx$ Integrating both sides: $\int \sin(v) dv = -\int \frac{1}{x} dx$ $-\cos(v) = -\log |x| + C$ $\cos(v) = \log |x| + C'$ Substituting $v = \frac{y}{x}$ back: $\cos\left(\frac{y}{x}ight) = \log |x| + C$.

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

Similar Questions

Let $f(x) = \sqrt{\lim_{r \rightarrow x} \left\{ \frac{2r^2 \left[(f(r))^2 - f(x)f(r)\right]}{r^2 - x^2} - r^3 e^{\frac{f(r)}{r}} \right\}}$ be differentiable in $(-\infty, 0) \cup (0, \infty)$ and $f(1) = 1$. Then the value of $ea$,such that $f(a) = 0$,is equal to:

$A$ bag contains $3$ red,$6$ white,and $7$ blue balls. Two balls are drawn one after another. What is the probability that the first ball is white and the second ball is blue,if the first ball drawn is not replaced in the bag?

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

In a single throw of two dice,what is the probability of obtaining a sum of numbers greater than $7$,given that $4$ appears on the first die?

Solve the differential equation $(x \,dy-y \,dx) y\, \sin \left(\frac{y}{x}\right)=(y \,dx+x\, dy) x\, \cos \left(\frac{y}{x}\right)$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module