Question

# Solve differential equation 2xy-9x^2+(2y+x^2+1)\frac{dy}{dx}=0, \ y(0)=-3

First order differential equations
Solve differential equation $$\displaystyle{2}{x}{y}-{9}{x}^{{2}}+{\left({2}{y}+{x}^{{2}}+{1}\right)}{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}={0},\ {y}{\left({0}\right)}=-{3}$$

2020-12-25
The equation cam be written as
$$\displaystyle{\left({2}{x}{y}-{9}{x}^{{2}}\right)}{\left.{d}{x}\right.}+{\left({2}{y}+{x}^{{2}}+{1}\right)}{\left.{d}{y}\right.}={0}$$
The equation is of the form
$$\displaystyle{M}{\left.{d}{x}\right.}+{N}{\left.{d}{y}\right.}={0}$$
$$\displaystyle{M}{y}={2}{x}$$
$$\displaystyle{N}{x}={2}{x}$$
The equation is exact since My=Nx
Assume that F(x,y) is the solution for the equation
$$\displaystyle{F}{x}={M}$$
$$\displaystyle={2}{x}{y}-{9}{x}^{{2}}$$
$$\displaystyle{F}=\int{2}{x}{y}-{9}{x}^{{2}}{\left.{d}{x}\right.}$$
$$\displaystyle{F}={x}^{{2}}{y}-{3}{x}^{{2}}+{g{{\left({y}\right)}}}$$
Take the derivative for F with respect to y
$$\displaystyle{F}{y}={x}^{{2}}+{g}'{\left({y}\right)}$$
$$\displaystyle{2}{y}+{x}^{{2}}+{1}={g}'{\left({y}\right)}$$
$$\displaystyle{g}'{\left({y}\right)}={2}{y}+{1}$$
$$\displaystyle{g}'{\left({y}\right)}=\int{2}{y}+{1}{\left.{d}{y}\right.}$$
$$\displaystyle{g}'{\left({y}\right)}={y}^{{2}}+{y}$$
Hence
$$\displaystyle{x}^{{2}}{y}-{3}{x}^{{3}}+{y}^{{2}}+{y}={C}$$
$$\displaystyle{y}^{{2}}+{\left({x}^{{2}}-{1}\right)}{y}-{3}{x}^{{2}}={C}$$
It is known that
$$\displaystyle{y}{\left({0}\right)}=-{3}$$
$$\displaystyle{9}-{3}={c}$$
$$\displaystyle{c}={6}$$
So the solution for the problem is
$$\displaystyle{y}^{{2}}{\left({x}^{{2}}-{1}\right)}{y}-{3}{x}^{{2}}-{6}={0}$$