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The general solution of the differential equation $2 x \frac{d y}{d x}-y=3$ is a family of
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parabolas
The given differential equation is
$2 x \frac{d y}{d x}-y=3$
$\Rightarrow \quad 2 x \frac{d y}{d x}=(y+3)$
$\Rightarrow \quad 2 \int \frac{\mathrm{dy}}{(\mathrm{y}+3)}=\int \frac{\mathrm{dx}}{\mathrm{x}}$ (on integrating)
$\Rightarrow \quad 2 \log (y+3)=\log x+\log c$
$\Rightarrow \quad(\mathrm{y}+3)^{2}=\mathrm{cx}$
which represents a family of parabolas.
$2 x \frac{d y}{d x}-y=3$
$\Rightarrow \quad 2 x \frac{d y}{d x}=(y+3)$
$\Rightarrow \quad 2 \int \frac{\mathrm{dy}}{(\mathrm{y}+3)}=\int \frac{\mathrm{dx}}{\mathrm{x}}$ (on integrating)
$\Rightarrow \quad 2 \log (y+3)=\log x+\log c$
$\Rightarrow \quad(\mathrm{y}+3)^{2}=\mathrm{cx}$
which represents a family of parabolas.
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