Learn how to solve problems step by step online. Solve the differential equation (sin(x)cos(x)-xy^2)dx+y(1-x^2)dy=0. The differential equation \left(\sin\left(x\right)\cos\left(x\right)-xy^2\right)dx+y\left(1-x^2\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 \frac{1}{2}\sin\left(x\right)^2-\frac{1}{2}y^2x^2 with respect to y to get.

Solve the differential equation (sin(x)cos(x)-xy^2)dx+y(1-x^2)dy=0