Question

Find the solution of the differential equation: $3x^2{y}^2+\cos \left(xy\right)-xy\sin \left(xy\right)+\frac{dy}{dx}(2x^{3}y-x^2\sin (xy\left)\right)=0$.

Answer 1

$[3x^2{y}^2+xy\sin (my\left)\right]dx+[2x^{3}y-x^2\sin (ny\left)\right]dy=0$

$\Rightarrow \mu (x,y)dx+N(x,y)dy=0$

$\frac{df}{dy}=\frac{dN}{dx}$

$\frac{df}{dx}=\mu (x,y)$

$\Rightarrow f(x,y)=\int \mu (x,y)dx+h\left(y\right)$

$\Rightarrow \int 3x^2{y}^{2}dx+\int \cos (x,y)dx-\int xy\sin \left(xy\right)dx+h\left(y\right)\Rightarrow x^3{y}^2+\frac{\sin \left(xy\right)}{y}+x\cos \left(yx\right)-\frac{\sin \left(xy\right)}{y}+h\left(y\right)\Rightarrow x^3{y}^2+x.\cos \left(yx\right)+h\left(y\right)\Rightarrow \frac{df}{dy}=x^3.2y-x^2\cos \left(xy\right)+\frac{dh\left(y\right)}{2}=N(x,y)$

$\Rightarrow \frac{dh\left(y\right)}{dy}=0\to h\left(y\right)=$ constant $=c$

$\Rightarrow f(x,y)=c$

$\Rightarrow x^3{y}^2+x\cos \left(xy\right)=c$