Question

Solve the equation (𝑥 + 𝑠𝑖𝑛𝑦)𝑑𝑥 + (𝑦2 + 𝑥𝑐𝑜𝑠𝑦)𝑑𝑦 = 0

Answer 1

$$\dfrac{\partial Q}{\partial x}=\cos y,\dfrac{\partial P}{\partial y}=\cos y$$

We have the following system of differential equations to find the function  $u(x, y)$

$$u=\int(x+\sin y)dx=\dfrac{x^2}{2}+x\sin y+\varphi(y)$$

$$\dfrac{\partial u}{\partial y}=x\cos y+\varphi'(y)$$

$$x\cos y+\varphi'(y)=y^2 +x\cos y$$

$$\varphi'(y)=y^2$$

$$\varphi(y)=\dfrac{y^3}{3}-C$$

So that the general solution of the exact differential equation is given by

$$\dfrac{x^2}{2}+x\sin y+\dfrac{y^3}{3}=C$$