Learn how to solve differential equations problems step by step online. Solve the differential equation cos(x)(1+cos(y))dx-sin(y)(1+sin(x))dy=0. The differential equation \cos\left(x\right)\left(1+\cos\left(y\right)\right)dx-\sin\left(y\right)\left(1+\sin\left(x\right)\right)dy=0 is exact, since it is written in the standard form M(x,y)dx+N(x,y)dy=0, where M(x,y) and N(x,y) are the partial derivatives of a two-variable function f(x,y) and they satisfy the test for exactness: \displaystyle\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}. In other words, their second partial derivatives are equal. The general solution of the differential equation is of the form f(x,y)=C. Using the test for exactness, we check that the differential equation is exact. Integrate M(x,y) with respect to x to get. Now take the partial derivative of \left(1+\cos\left(y\right)\right)\sin\left(x\right) with respect to y to get.

Solve the differential equation cos(x)(1+cos(y))dx-sin(y)(1+sin(x))dy=0