Question

(xy2-x2)dx+(3x2y2+x2y-2x3+y2)dy=0​

Answer 1

Answer:

xy2-x2dx+3x2y2+x2y-2x3+2y2dy=0

Answer 2

The equation is $e^6^y ( x^2y^2/2 - x^3/3+18y^2-6y+1/108)=C$

Given:

Derivative of x = (xy2-x2)

Derivative of y = (3x2y2+x2y-2x3+y2)

To find:

Solve (xy2-x2)dx+(3x2y2+x2y-2x3+y2)dy=0​

Solution:

The differential calculation is based on the derivatives of the following algebraic equations.

$A = xy^2 - x^2$

$B = 3x^2y^2 + x^2y-2x^3+y^2$

Let us differentiate the following equations separately,

$dA/dy = 2xy$

$dB/dx = 6xy^2 + 2xy - 6x^2$

We can see that both the differential equations are not equal to each other. Which means that the equations are not exact in nature.

So if,

$1/A (dB/dx - dA/dx)$ is in y only the integrating factor would be,

$e^\int\limits{^1^/^A ^(^d^B^/^d^x ^- ^d^A^/^d^y^)} \, ^d^y^$  

So according to the concept,

$1/A [dB/dx - dA/dy] = 6xy^2 - 6x^2/ xy^2 - x^2$

$6 ( xy^2 - x^2/xy^2 - x^2)$

$= 6$

The six can be also written as $6y^o$ since $y^o$ is 1.

the integrating factor would be,

$e^\int\limits ^{6y^0} \, ^d^y$

$e^6^y$

Now multiply the equation with the integrating factor,

$e^6^y(xy^2-x^2) dx - (x^2y^2 + x^2y-2x^3+y^2)e^6^ydy=0$

Now we can call the equation an exact equation

Now integrate on both sides,

$g(y) = \int\limits {(e^6^yy^2)} \, dy = y^2e^6y/6 - \int\limits{2e^6^y/6} \, dy$

$y^2e^6^y/6-ye^6^y/18-e^6^y/108 = e^6^y(18y^2=6y+1)/108$

$e^6^y(x^2y^2/2 - x^3/3) + e^6^y(18y^2-6y+1)/108$

The general solution can be written as,

$e^6^y ( x^2y^2/2 - x^3/3+18y^2-6y+1/108)=C$

#SPJ6