Question

Find the general/particular solution of the following Differential Equations

(Non-Exact D.E)

(2x² - 2y² + 2xy)dx + (x² - 2y)dy=0

Answer 1

Solution;

$M=2x^2-2y^2+2xy$

$\frac{dM}{dy}=-4y+2x=2x-4y$

$N=x^2-2y$

$\frac{dN}{dx}=2x$

Clearly;

$\frac{dM}{dy}\neq\frac{dN}{dx}$

The equation is not exact.

Use $e^{2x}$ as the integrating factor to make the equation exact.

The solution will be ;

$U(x,y)=\int e^{2x}(x^2-2y)dy$

$U(x,y)=e^{2x}x^2\int1dy-2e^{2x}\int ydy$

$U(x,y)=x^2e^{2x}y-e^{2x}y^2+C$