Find a solution for the differential equation dy/dx=exp(-y)*sin2x which passes through the origin.

First separate the variables so that the left side of the equation is an expression only in terms of y and the right side only in terms of x.exp(y)dy=sin2x dxSecondly both sides have to be integrated to obtain an expression for exp(y) in terms of x.exp(y)=-(1/2)cos2x+cFinally take the natural log to find an expression for y in terms of x, the general solution.y=ln(-(1/2)cos2x+c)To work out the constant c substitute the x and y coordinates of a known point into the general solution. In this case the origin(x=0,y=0).0=ln(-(1/2)+c) Therefore c=3/2 and the particular solution to the differential equation is y=ln(-(1/2)cos2x+3/2)y=ln((3-cos2x)/2)