Final answer:
To find the general solution to the given equation, we can use the method of exact differential equations. After verifying that the equation is exact, we can find the potential function ϕ(x, y) and solve for y in terms of x to obtain the general solution.
Step-by-step explanation:
To find the general solution to the given equation, (2xy+e⁻ʸ+4x)dx = (3cosh(y)+xe⁻ʸ-x²)dy, we can use the method of exact differential equations. To do this, we need to check whether the equation is exact by verifying if the partial derivatives of the left-hand side and right-hand side satisfy the condition ∂M/∂y = ∂N/∂x. If the equation is exact, we can find a potential function ϕ(x, y) such that ∂ϕ/∂x = M and ∂ϕ/∂y = N, where M and N are the coefficients of dx and dy, respectively.
After verifying that the equation is exact, we can find the potential function ϕ(x, y) by integrating the coefficient of dx with respect to x, treating y as a constant, and integrating the coefficient of dy with respect to y, treating x as a constant. This will give us the general solution ϕ(x, y) = C, where C is the constant of integration.
Finally, we can solve for y in terms of x to obtain the general solution to the original equation.