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• Notes

$\left(y^{2}+y x\right) d x-x^{2} d y=0$

$\left(M(x, y) d x_{+} N(x, y) d y=0\right.$ homog

$u=f(x) \quad y=u x$

$d y=x d u+u d x$

$\left(u^{2} x^{2}+u x^{2}\right) d x-x^{2}(x d u+u d x)=0$

$u^{2} x^{2} d x+u x^{2} d x-x^{3} d u-x^{2} u d x=0$

$u^{2} x^{2} d x-x^{3} d u=0$

$\div u^{2} x^{3}$

$\frac{u^{2} x^{2}}{u^{2} x^{3}} d x-\frac{x^{3}}{u^{2} x^{3}} d u=0$

$\frac{d x}{x}-\frac{d u}{u^{2}}=0$

$\frac{d x}{x}=\frac{d u}{u^{2}}$

$\int \frac{d x}{x}=\ln |x|$

$\int u^{-2} d u=-u^{-1}$

$\ln |x|=-u^{-1}+C$

$y=u x \quad u=\frac{y}{x} \quad u^{-1}=\frac{x}{y}$

$\ln |x|=-\frac{x}{y}+c$

$y \ln |x|+x=c y$

$\left(y^{2}+y x\right) d x-x^{2} d y=0$

$V=f(y) \quad x=v y \quad d x=v d y+y d v$

$\left(y^{2}+v y^{2}\right)(v d y+y d v)-v^{2} y^{2} d y=0$

$y^{2} v d y+y^{3} d v+v^{2} y^{2} d y+v y^{3} d v_{-} v^{2} y^{2} d y = 0$

$y^{2} v d y+\left(y^{3}+v y^{3}\right) d v=0$

$\div V y^{3}$

$\frac{y^{2} v}{v y^{3}} d y+\left(\frac{y^{3}}{v y^{3}}+\frac{v y^{3}}{v y^{3}}\right) d v=0$

$\frac{d y}{y}+\left(\frac{1}{v}+1\right) d v=0$

$\frac{d y}{y}=-\left(\frac{1}{v}+1\right) d v$

$\ln |y|=-\ln |v|-v+c$

$v=\frac{x}{y}$

$\ln |y|=-\ln |x / y|-x / y+c$

$=-[\ln |x|-\ln |y|]-x / y+c$

$\ln |y|=-\ln |x|+\ln |y|-x / y+c$

$\ln |x|+x / y=c$

$y \ln |x|+x=c y$

$y d x+x(\ln x-\ln y-1) d y=0 \quad y(1)=e$

$v=f(y) \quad x=v y \quad d x=v d y+y d v$

$y(v d y+y d v)+v y(\ln (v y)-\ln y-1) d y=0$

$\ln v y=\ln v+\ln y$

$y v d y+y^{2} d v+v y(\ln v+\ln y-\ln y-1) d y=0$

$y v d y+y^{2} d v+v y \ln v d y-v y d y=0$

$y^{2} d v+v y \ln v d y=0$

$\div y^{2} v \ln v$

$\frac{y^{2}}{y^{2} v \ln v} d v+\frac{v y \ln v}{y^{2} v \ln v} d y=0$

$\frac{d v}{v \ln v}+\frac{d y}{y}=0$

$\frac{d v}{v \ln v}=-\frac{d y}{y}$

$\int \frac{(1 / v) dv}{\ln v}=\ln (\ln v)$

$\ln |\ln | v| |=-\ln |y|+c$

$\ln |\ln | x / y| |=-\ln |y|+c$

$e^{{\ln |\ln | x / y} | |}=e^{-\ln |y|+c}$

$\ln \left|x \right/ y |=e^{c-\ln |y|}=\frac{e^{c}}{e^{\ln |y|}}$

$\ln \left|x / y\right|=\frac{e^{c}}{y}$

$y \ln \left|x / y\right|=C_{1}$

$C_{1}=e^{c}$

$C_{1}=e \ln \left|\frac{1}{e}\right|$

$C_{1}=-e$

$y \ln |x/y |=-e$