Differential equations are solved by reducing them to the exact differential of an expression in
x x x and
y y y i.e., they are reduced to the form
d ( f ( x , y ) ) = 0 d(f(x,y)) = 0 d ( f ( x , y )) = 0 .
e.g.:
x d x + y d y x 2 + y 2 = y d x − x d y x 2 \frac{x\,dx + y\,dy}{\sqrt{x^2 + y^2}} = \frac{y\,dx - x\,dy}{x^2} x 2 + y 2 x d x + y d y = x 2 y d x − x d y ⇒ 1 2 2 x d x + 2 y d y x 2 + y 2 = − x d y − y d x x 2 ⇒ d ( x 2 + y 2 ) = − d ( y x ) \Rightarrow \frac{1}{2}\frac{2x\,dx + 2y\,dy}{\sqrt{x^2 + y^2}} = -\frac{x\,dy - y\,dx}{x^2} \Rightarrow d\left(\sqrt{x^2 + y^2}\right) = -d\left(\frac{y}{x}\right) ⇒ 2 1 x 2 + y 2 2 x d x + 2 y d y = − x 2 x d y − y d x ⇒ d ( x 2 + y 2 ) = − d ( x y ) ⇒ d ( x 2 + y 2 + y x ) = 0 \Rightarrow d\left(\sqrt{x^2 + y^2} + \frac{y}{x}\right) = 0 ⇒ d ( x 2 + y 2 + x y ) = 0 ∴ \therefore ∴ solution is
x 2 + y 2 + y x = c \sqrt{x^2 + y^2} + \frac{y}{x} = c x 2 + y 2 + x y = c Use the above method to answer the following question:
The general solution of
( 2 x 3 − x y 2 ) d x + ( 2 y 3 − x 2 y ) d y = 0 (2x^3 - xy^2)\,dx + (2y^3 - x^2y)\,dy = 0 ( 2 x 3 − x y 2 ) d x + ( 2 y 3 − x 2 y ) d y = 0 is