$1.$ Find the differential equation of one$-$parameter families y $=$ cos$^2($x $+$ c$).(25$ points$)2.$ Solve the Initial value problem, $(2$xy$)*$ dx $+(3$x $^2+4$y$)*$ dy $=0$ ; y$(1)=-2(25$ points$)3.$ Find the general solution of the following equation, $(6$x $+4$y $+1)*$ dx $+(4$x $+2$y $+2)*$ dy $=0(25$ points$)4.$ A given family of curves is said to be self$-$orthogonal if its family of orthogonal trajectories is the same as the given family. Show that the family of parabolas,y $^2=2$cx $+$ c $^2$ is self$-$orthogonal. $(25$ points