2. y sin 2x dx - (1 + y² + cos² x) dy = 0

This differential equation is of the form M ( x , y ) d x + N ( x , y ) d y = 0 , where M ( x , y ) = y sin 2 x and N ( x , y ) = − ( 1 + y 2 + cos 2 x ) .
To solve this differential equation, we need to check whether it is exact. A differential equation is exact if the following condition is met:
∂ y ∂ M ​ = ∂ x ∂ N ​
Let's calculate these partial derivatives:

∂ y ∂ M ​ = ∂ y ∂ ​ ( y sin 2 x ) = sin 2 x

∂ x ∂ N ​ = ∂ x ∂ ​ ( − ( 1 + y 2 + cos 2 x )) = − 2 cos x sin x


The derivatives sin 2 x and − 2 cos x sin x do not match, which means the differential equation is not exact.
In cases where the differential equation is not exact, we might need to find an integrating factor, or use another method to solve it based on the particular nature of M and N , or any existing symmetries or substitutions that might simplify the equation.
One typical method to attempt is to divide through either by M ( x , y ) or N ( x , y ) and try to identify if this is a separable, homogeneous, or can be converted into a linear differential equation.
Without further simplifying assumptions or substitutions, more information or context might be needed to proceed further with solving the equation.

Answered by BenjaminOwenLewis | 2025-07-21