MCQ
વિકલ સમીકરણ $(x + y)dx + xdy = 0$ નો ઉકેલ મેળવો.
- A${x^2} + {y^2} = c$
- B$2{x^2} - {y^2} = c$
- ✓${x^2} + 2xy = c$
- D${y^2} + 2xy = c$
==> $\frac{{dy}}{{dx}} = - \frac{{x + y}}{x}$
It is homogenous equation, hence put $y = vx$ and $\frac{{dy}}{{dx}} = v + x\frac{{dv}}{{dx}},$ we get $v + x\frac{{dv}}{{dx}} = - \frac{{x + vx}}{x} = - \frac{{1 + v}}{1}$
==> $x\frac{{dv}}{{dx}} = - 1 - 2v$==> $\int_{}^{} {\frac{{dv}}{{1 + 2v}}} = - \int_{}^{} {\frac{{dx}}{x}} $
==> $\frac{1}{2}\log (1 + 2v) = - \log x + \log c$ ==> $\log \left( {1 + 2\frac{y}{x}} \right) = 2\log \frac{c}{x}$
==> $\frac{{x + 2y}}{x} = {\left( {\frac{c}{x}} \right)^2}$ ==> ${x^2} + 2xy = c$.
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