Learn how to solve problems step by step online. Solve the differential equation (cos(x)-xsin(x)y^2)dx+2xydy=0. The differential equation \left(\cos\left(x\right)-x\sin\left(x\right)+y^2\right)dx+2xy\cdot 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 x\cos\left(x\right)+y^2x with respect to y to get.

Solve the differential equation (cos(x)-xsin(x)y^2)dx+2xydy=0