Answer to Question #320862 in Differential Equations for Tin Jus

Question

Use any appropriate method in finding the indicated solution of the following differential 

equations. 

[2𝑥𝑦 cos(𝑥2) − 2𝑥𝑦 + 1]𝑑𝑥 + [𝑠𝑖𝑛(𝑥2) − 𝑥2]𝑑𝑦 = 0

Answer 1

$\left[ 2xy\cos x^2-2xy+1 \right] dx+\left[ \sin x^2-x^2 \right] dy=0\\\frac{\partial}{\partial x}\left[ y\sin x^2-x^2y+x \right] dx+\frac{\partial}{\partial y}\left[ y\sin x^2-x^2y+x \right] dy=0\\d\left[ y\sin x^2-x^2y+x \right] =0\\y\sin x^2-x^2y+x=C\\y=\frac{C-x}{\sin x^2-x^2}$