Solved: (a) Show that a one-parameter family of solutions of the equation (4xy 3x2 ) dx

Chapter 2, Problem 39

(choose chapter or problem)

(a) Show that a one-parameter family of solutions of the equation

\(\left(4 x y+3 x^{2}\right) d x+\left(2 y+2 x^{2}\right) d y=0\)

is \(x^{3}+2 x^{2} y+y^{2}=c\).

(b) Show that the initial conditions y(0) = -2 and y(1) = 1 determine the same implicit solution.

(c) Find explicit solutions \(y_{1}(x)\) and \(y_{2}(x)\) of the differential equation in part (a) such that \(y_{1}(0)=-2\) and \(y_{2}(1)=1\). Use a graphing utility to graph \(y_{1}(x)\) and \(y_{2}(x)\)).