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Question Number 180801 by Mastermind last updated on 17/Nov/22

$$\mathrm{Solve}\mathrm{the}\mathrm{Differential}\mathrm{equation}:$$$$(3\mathrm{xy}+{6\mathrm{y}}^2)\mathrm{dx}+({2\mathrm{x}}^2+9\mathrm{xy})\mathrm{dy}=0$$$$$$$$\mathrm{Mastermind}$$

Answered by qaz last updated on 17/Nov/22

$$\frac{dy}{dx}=-\frac{3xy+6y^2}{2x^2+9xy}=-\frac{3\frac{y}{x}+6{\left(\frac{y}{x}\right)}^2}{2+9\frac{y}{x}}$$$$z=\frac{y}{x}$$$$\frac{dy}{dx}=\frac{d\left(xz\right)}{dx}=\frac{zdx+xdz}{dx}=-\frac{3z+6z^2}{2+9z}$$$$\Rightarrow \frac{2+9z}{5z+15z^2}dz=-\frac{dx}{x}$$$$\Rightarrow \frac{2}{5}lnz+\frac{1}{5}ln(3z+1)+C=-lnx$$$$\Rightarrow 3x^2{y}^3+x^3{y}^2=C$$

Commented by Mastermind last updated on 17/Nov/22

$$\mathrm{Thanks}$$

Commented by SLVR last updated on 18/Nov/22

$$nicesir..$$

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