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Question Number 118954 by bramlexs22 last updated on 21/Oct/20

$$solve(6x^2+3y^2)dx=2xydy$$

Answered by john santu last updated on 21/Oct/20

$$sety=gx\Rightarrow \frac{dy}{dx}=g+x\frac{dg}{dx}$$$$thedifferentialequationcanbewe$$$$writteas\frac{dy}{dx}=\frac{6x^2+3y^2}{2xy}$$$$\Rightarrow g+x\frac{dg}{dx}=\frac{6x^2+3g^2{x}^2}{2{gx}^2}=\frac{6+3g^2}{2g}$$$$\Rightarrow x\frac{dg}{dx}=\frac{6+g^2}{2g};\frac{2gdg}{6+g^2}=\frac{dx}{x}$$$$\Rightarrow \int \frac{2g}{6+g^2}-\int \frac{dx}{x}=c$$$$\Rightarrow \ln \left(\frac{6+g^2}{x}\right)=c\Rightarrow \frac{6+\left(\frac{y^2}{x^2}\right)}{x}=C$$$$\Rightarrow \frac{6x^2+y^2}{x^3}=C$$

Answered by 1549442205PVT last updated on 21/Oct/20

$$\mathrm{Put}\mathrm{y}=\mathrm{tx}\Rightarrow \mathrm{dy}=\mathrm{xdt}+\mathrm{tdx}$$$$({6\mathrm{x}}^2+{3\mathrm{y}}^2)\mathrm{dx}=2\mathrm{xydy}$$$$\iff ({6\mathrm{x}}^2+{3\mathrm{x}}^2{\mathrm{t}}^2)\mathrm{dx}={2\mathrm{x}}^2\mathrm{t}(\mathrm{xdt}+\mathrm{tdx})$$$$\iff (6+{3\mathrm{t}}^2)\mathrm{dx}=2\mathrm{t}(\mathrm{xdt}+\mathrm{tdx})$$$$\iff ({\mathrm{t}}^2+6)\mathrm{dx}=2\mathrm{xtdt}\Rightarrow \frac{\mathrm{dx}}{\mathrm{x}}=\frac{2\mathrm{tdt}}{{\mathrm{t}}^2+6}$$$$\iff \ln \mid \mathrm{x}\mid +\mathrm{lnC}=\ln ({\mathrm{t}}^2+6)$$$$\iff \mathrm{lnC}\mid \mathrm{x}\mid =\ln ({\mathrm{t}}^2+6)\iff \mathrm{C}\mid \mathrm{x}\mid ={\mathrm{t}}^2+6$$$$\iff \mathrm{C}\mid \mathrm{x}\mid =\frac{{\mathrm{y}}^2}{{\mathrm{x}}^2}+6\iff {\mathrm{y}}^2={\mathrm{Cx}}^2\mid \mathrm{x}\mid -{6\mathrm{x}}^2$$$$(\mathrm{C}>0-\mathrm{constant})$$

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