Answer to Question #231184 in Differential Equations for Randal Rodriguez

Question

eliminate the arbitrary constant using any method

13.x^2y^3\:+\:x^3y^5\:=\:C

Answer 1

$x^2y^3+x^3y^5=c$

Differentiating both sides w.r.t $x$ , we get

$\frac{d}{dx}(x^2y^3+x^3y^5)=\frac{d}{dx}(c) \\\Rightarrow \frac{d}{dx}(x^2y^3)+\frac{d}{dx}(x^3y^5)=0 \\\Rightarrow 2xy^3+x^2.3y^2\frac{dy}{dx}+3x^2y^5+x^3.5y^4\frac{dy}{dx}=0 \\\Rightarrow \frac{dy}{dx}(3x^2y^2+5x^3y^4)+2xy^3+3x^2y^5=0 \\\Rightarrow \frac{dy}{dx}=-\frac{2xy^3+3x^2y^5}{3x^2y^2+5x^3y^4}$