$2\mathrm{.}4$ Exercises $($pp$.66-68)$

$3.$ Determine whether the given differential equation is exact. If it is exact, solve it$.$

$(1)(2y-\frac{1}{x}+cos3x)(\frac{dy}{dx})+\frac{y}{x^2}-4x^3+3ysin3x=0.$

$(2)(ylny-e^{-xy})dx+(\frac{1}{y}+xlny)dy=0.$

$(3)x\frac{dy}{dx}=2x{e}^x-y+6x^2.$

$(4)(\frac{1}{t}+\frac{1}{t^2}-\frac{y}{t^2+y^2})dt+(y{e}^y+\frac{1}{t^2+y^2})dy=0.$

$4.$ Solve the given initial$-$value problem.

$(1)(x+y{)}^{2}dx+(2xy+x^2-1)dy=0,y(1)=1.$

$(2)(e^x+y)dx+(2+x+y{e}^2)dy=0,y(0)=1.$

$(3)(\frac{1}{1+y^2}+cosx-2xy)(\frac{dy}{dx})=y(y+sinx),y(0)=1.$