Kaycee Roche

2020-12-24

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

Neelam Wainwright

Skilled2020-12-25Added 102 answers

The equation cam be written as

$(2xy-9{x}^{2})dx+(2y+{x}^{2}+1)dy=0$

The equation is of the form

$Mdx+Ndy=0$

$My=2x$

$Nx=2x$

The equation is exact since My=Nx

Assume that F(x,y) is the solution for the equation

$Fx=M$

$=2xy-9{x}^{2}$

$F=\int 2xy-9{x}^{2}dx$

$F={x}^{2}y-3{x}^{2}+g\left(y\right)$

Take the derivative for F with respect to y

$Fy={x}^{2}+{g}^{\prime}\left(y\right)$

$2y+{x}^{2}+1={g}^{\prime}\left(y\right)$

${g}^{\prime}\left(y\right)=2y+1$

${g}^{\prime}\left(y\right)=\int 2y+1dy$

${g}^{\prime}\left(y\right)={y}^{2}+y$

Hence

${x}^{2}y-3{x}^{3}+{y}^{2}+y=C$

${y}^{2}+({x}^{2}-1)y-3{x}^{2}=C$

It is known that

$y\left(0\right)=-3$

$9-3=c$

$c=6$

So the solution for the problem is

${y}^{2}({x}^{2}-1)y-3{x}^{2}-6=0$

The equation is of the form

The equation is exact since My=Nx

Assume that F(x,y) is the solution for the equation

Take the derivative for F with respect to y

Hence

It is known that

So the solution for the problem is

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