Question

1.(x^2-yx^2dy/dx+y^2+xy^2=0 2.sec^2xtanydx+sec^2ytanxdy=0 3.(y/xdy/dx=(1+x^2+y^2+x^2y^2^1/2 4.dy/dx=xe^(y-x^2,if y=0 when x=0 5.dy/dx=e^(2x-3y+4x^2e^-3y 6.(x+1dy/dx+1=2e^-y 7.(x-y^2dy/dx=a^2 8.dy/dx=cos(x+y+1

Answer 1

$(x^2-yx^2)\dfrac{\mathrm dy}{\mathrm dx}+y^2+xy^2=0\implies \dfrac{y-1}{y^2}\,\mathrm dy=\dfrac{1+x}{x^2}\,\mathrm dx$

Integrating both sides yields

$\displaystyle\int\frac{y-1}{y^2}\,\mathrm dy=\int\frac{1+x}{x^2}\,\mathrm dx$
$\displaystyle\int\left(\frac1y-\frac1{y^2}\right)\,\mathrm dy=\int\left(\frac1x+\frac1{x^2}\right)\,\mathrm dx$
$\ln|y|+\dfrac1y=\ln|x|-\dfrac1x+C$