Test for exactness. If exact, solve. If not, use an integrating factor as given or obtained by inspection or by the theorems in the text. Also, if an initial condition is given, find the corresponding particular solution.

$2xydx+x^{2}dy=0$

$x^{3}dx+y^{3}dy=0$

$sinxcosydx+cosxsinydy=0$

$e^3(dr+3rd)=0$

$(x^2+y^2)dx-2xydy=0$

$3(y+1)dx=2xdy,(y+1)x^{-4}$

$2xtanydx+se{c}^{2}ydy=$

ODEs. INTEGRATING FACTORS

Test for exactness. If exact, solve. If not, use an integrating factor as given or obtained by inspection or by the theorems in the text. Also, if an initial condition is given, find the corresponding particular solution.

$2xydx+x^{2}dy=0$

$x^{3}dx+y^{3}dy=0$

$sinxcosydx+cosxsinydy=0$

$e^3(dr+3rd)=0$

$(x^2+y^2)dx-2xydy=0$

$3(y+1)dx=2xdy,(y+1)x^{-4}$

$2xtanydx+se{c}^{2}ydy=0$

$e^x(cosydx-sinydy)=0$

$e^{2x}(2cosydx-sinydy)=0,y(0)=0$

$ydx+[y+tan(x+y)]dy=0,cos(x+y)$

$2coshxcosydx=sinhxsinydy$

$(2xydx+dy)e^{x^2}=0,y(0)=2$

$e^{-y}dx+e^{-x}(-e^{-y}+1)dy=0,F=e^{x+y}$

$(a+1)ydx+(b+1)xdy=0,y(1)=1,F=x^a{y}^b$

Exactness. Under what conditions for the constants $a,b,k,l$ is $(ax+by)dx+(kx+ly)dy=0$ exact? Solve the exact ODE.