TS EAMCET · Maths · Differential Equations
The general solution of the differential equation \(\left(x^3-y^3\right) d x=\left(x^2 y-x y^2\right) d y\) is
- A \(y=x \log (\mathrm{c}|x+y|)\)
- B \(y=\log (\mathrm{c}|x+y|)\)
- C \(x y=\log (\mathrm{c}|x+y|)\)
- D \(x+y+\log |x+y|+\mathrm{c}=0\)
Answer & Solution
Correct Answer
(A) \(y=x \log (\mathrm{c}|x+y|)\)
Step-by-step Solution
Detailed explanation
Substitute \(y=vx \Rightarrow \frac{dy}{dx}=v+x\frac{dv}{dx}\): \(v+x\frac{dv}{dx} = \frac{x^3-(vx)^3}{x^2(vx)-x(vx)^2}\) \(x\frac{dv}{dx} = \frac{1-v^3}{v-v^2}-v = \frac{1-v^2}{v-v^2} = \frac{1+v}{v}\) \(\frac{v}{1+v}dv = \frac{1}{x}dx\)…
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