The solutions of the differential equation dydx=siny+xsin2y−xcosy is

The solution to the differential equation dy/dx = sin(y) + xsin(2y) - xcos(y) cannot be expressed in terms of elementary functions. This is because the equation is non-linear and does not have a constant coefficient.

However, you can use numerical methods to approximate the solution. Here is a general approach:

1. Rewrite the equation in the form dy/dx = f(x, y), where f(x, y) = sin(y) + xsin(2y) - xcos(y).

2. Choose an initial condition, say (x0, y0). This is the point where you start the approximation.

3. Choose a step size h. This is the distance you move along the x-axis for each step.

4. Calculate the slope at the initial point using the function f(x, y). This is the value of dy/dx at the initial point.

5. Use the slope to estimate the value of y at x0 + h. This is done using the formula y1 = y0 + h*f(x0, y0).

6. Repeat steps 4 and 5 for the new point (x1, y1) to get the next point (x2, y2), and so on.

This method is called Euler's method. It is a simple but not very accurate method. There are more accurate methods like Runge-Kutta methods.

Please note that these methods only give approximate solutions. The accuracy of the solution depends on the step size h. The smaller the step size, the more accurate the solution. However, a smaller step size requires more computations.