AP EAMCET · Maths · Differential Equations
The general solution of the differential equation \(\left(x \sin \frac{y}{x}\right) d y=\left(y \sin \frac{y}{x}-x\right) d x\) is
- A \(\cos \left(\frac{y}{x}\right)=\log |x|+c\)
- B \(\cos \left(\frac{y}{x}\right)=\frac{1}{x}+c\)
- C \(\cos \left(\frac{x}{y}\right)=\log |y|+c\)
- D \(\cos \frac{\mathrm{y}}{\mathrm{x}}=\frac{2}{\mathrm{x}}+\mathrm{c}\)
Answer & Solution
Correct Answer
(A) \(\cos \left(\frac{y}{x}\right)=\log |x|+c\)
Step-by-step Solution
Detailed explanation
\(\frac{dy}{dx} = \frac{y \sin \frac{y}{x} - x}{x \sin \frac{y}{x}} = \frac{y}{x} - \frac{1}{\sin \frac{y}{x}}\) Let \(y=vx \Rightarrow \frac{dy}{dx}=v+x\frac{dv}{dx}\). \(v+x\frac{dv}{dx} = v - \frac{1}{\sin v}\) \(x\frac{dv}{dx} = -\frac{1}{\sin v}\)…
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