differential equation

Question

Solve the differential equation (x2 + 3y2)dx 2xydy = 0 with initial condition y(2) = 6.

Answer 1

x²+3y²+2xydy/dx=0 ... is of the form M(x,y)+N(x,y)dy/dx=0 ... could be an "exact" equation

My=6y and Nx=2y indicates that this is not an exact DE and will need an Integrating factor.

Let (My-Nx)/N = f(x) = (6y-2y)/2xy=2/x and the IF = e^{integral(2/x)}=x²

Define new problem to be x²(x²+3y²)+2xyx²=0 M=x²(x²+2y²) and N=2yx³ , an exact DE

Note that F= integral Mdx = x⁵/5+ 2yx³ +h(y)

Note that Fy=2yx³+ h'=N=2yx³ thus h'=0 and h = constant

Note that x⁵/5+ 2yx³=constant is a solution