Question

(𝒚 𝟐 + 𝒙𝒚𝟑 ) 𝒅𝒙 + (𝟓𝒚 𝟐 − 𝒙𝒚 + 𝒚 𝟐 𝐬𝐢𝐧 𝒚) 𝒅𝒚 = 𝟎 

Answer 1

Given equation is of the form- $Mdx+Ndy=0$

where, $M(x,y)=y^2+xy^3\Rightarrow M_y=2y+3xy^2$

$N(x,y)=5y^2-xy+y^2siny\Rightarrow N_x=-y$

So, $\dfrac{N_x-M_y}{M}=\dfrac{-y-(2y+3xy^2)}{y^2+xy^3}=\dfrac{-3y(1+xy)}{y^2(1+y)}=\dfrac{-3}{y}$

Integrating Factor $\mu=e^{-\int\frac{3}{y}dy}=e^{-3lny}=y^{-3}$

So Its, solution is-

$(\dfrac{1}{y}+x)dx+(\dfrac{5}{y}-\dfrac{x}{y^2}+siny)dy=0$

$f_x=\dfrac{1}{y}+x, f_y=\dfrac{5}{y}-\dfrac{x}{y^2}+siny$

and ,

$F(x,y)=F_x+F_y=\dfrac{x^2}{2}+\dfrac{x}{y}+5lny-cosy$

So The complete solution is-

$F(x,y)=c$

$\Rightarrow \dfrac{x^2}{2}+\dfrac{x}{y}+5lny-cosy=c$