Solve for solution xcosydy + sin ydx= xsin^3ydx

Answer 1

This differential equation can be separated -- this means that you can manipulate the terms around so that you can get one side of the equation only involving dx and the other side only involving dy. Once you do this, you can then integrate each side.......

cos(y) dy = sin³(y) dx - x^-1 sin(y) dx

∫ cos(y) dy = ∫ sin³(y) dx - x^-1 ∫ sin(y) dx

The left side of the equation is an integration over y, so x is considered a constant. Likewise, the right side of the equation is an integration over x, so y is considered a constant. Given this, the left side of the equation integrates to:

sin(y) + f(x)

Where f(x) is an arbitrary function of x.

Integrating the right side of the equation with respect to x:

x sin³(y) - sin(y) ln(x) + g(y)

Where g(y) is an arbitrary function of y.

The solution can then be written as:

sin(y) + f(x) = x sin³(y) - sin(y) ln(x) + g(y)