Respuesta :
Answer with explanation:
⇒(2 x y )d x+(x-6 y) d y=0
P= 2 x y
Q=x-6 y
[tex]P_{y}=2 x\\\\Q_{x}=1[/tex]
So this Differential Equation is exact.
To solve this, we will first evaluate,[tex]\varphi (x,y)[/tex].
[tex]\varphi_{x}=P\\\\\varphi_{y}=Q\\\\\varphi=\int P d x\\\\= \int 2 x y dx\\\\\varphi=x^2 y\\\\\varphi(x,y)=x^2y+k(y)------(1)[/tex]
Differentiating with respect to , y
[tex]\varphi'(x,y)=x^2+k'(y)=Q=x-6 y\\\\\rightarrow x-6 y-x^2=k'(y)\\\\ k(y)=\int (x-6 y -x^2) dy\\\\k(y)=x y-3 y^2-x^2 y+f\\\\\varphi(x,y)=x^2y+x y-3 y^2-x^2 y+f\\\\\varphi(x,y)=x y-3 y^2+f[/tex]
Substituting the value of , k(y) in equation 1.
This is required Solution of exact differential equation.
