Q119 / 160 · AP EAMCET 2022 (06 Jul Shift 2) · Mathematics

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

  • A
    2
  • Correct answer
    B
    1
  • C
    -1
  • D
    -2

Solution

$$ \begin{aligned} & \text { }\left(x \sin \frac{y}{x}\right) d y=\left(y \sin \frac{y}{x}-x\right) d x \\ & \frac{d y}{d x}=\frac{y \sin y / x-x}{x \sin \frac{y}{x}} \Rightarrow \frac{d y}{d x}=\frac{y}{x}-\frac{1}{\sin \frac{y}{x}} \\ & \text { Let } \frac{y}{x}=v \text { or } y=v x \Rightarrow \frac{d y}{d x}=v+x \frac{d v}{d x} \\ & \Rightarrow v+x \frac{d v}{d x}=v-\frac{1}{\sin v} \Rightarrow-\int \sin v d v=\int \frac{1}{x} d x \\ & \Rightarrow \cos v=\log x+C \Rightarrow \cos \frac{y}{x}=\log x+C \end{aligned} $$ Compare it with given equation, we get $A=1$

Hence, the correct answer is option: 'B'.

Relevant Questions