MCQ
નીચેનામાંથી કયું વિકલ સમીકરણ સમપરિમાણ છે ?
- A$(4 x+6 y+5) d y-(3 y+2 x+4) d x=0$
- B$(x y) d x-\left(x^{3}+y^{3}\right) d y=0$
- C$\left(x^{3}+2 y^{2}\right) d x+2 x y d y=0$
- D$y^{2} d x+\left(x^{2}-x y^{2}-y^{2}\right) d y=0$
Consider the equation given in alternative $D$ :
$y^{2} d x+\left(x^{2}-x y^{2}-y^{2}\right) d y=0$
$\Rightarrow \frac{d x}{d y}=\frac{-y^{2}}{x^{2}-x y^{2}-y^{2}}=\frac{y^{2}}{y^{2}+x y-x^{2}}$
Let $F(x,y) = \frac{{{y^2}}}{{{y^2} + xy - {x^2}}}$
$\Rightarrow \mathrm{F}(\lambda \mathrm{x}, \lambda \mathrm{y})=\frac{(\lambda \mathrm{y})^{2}}{(\lambda \mathrm{y})^{2}+(\lambda \mathrm{x})(\lambda \mathrm{y})-(\lambda \mathrm{x})^{2}}$
$=\frac{\lambda^{2} \mathrm{y}^{2}}{\lambda^{2}\left(\mathrm{y}^{2}+\mathrm{xy}-\mathrm{x}^{2}\right)}$
$=\lambda^{o}\left(\frac{\mathrm{y}^{2}}{\mathrm{y}^{2}+\mathrm{xy}-\mathrm{x}^{2}}\right)$
$=\lambda^{o} \mathrm{F}(\mathrm{x}, \mathrm{y})$
Hence, the differential equation given in alternative $\mathrm{D}$ is a homogenous equation.
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