Final answer:
The question seems to contain a typographical error and requires clarification on the differential equation's correct form. Depending on whether the equation is dx/dy or dy/dx, methods such as separation of variables or the integrating factor can be utilized to find the general solution.
Step-by-step explanation:
The question requires finding the general solution to a differential equation of the form dx/dy + 9y = 1/ cos3x −15x+7+e*x. This equation appears to be incorrectly typed as dx/dy usually indicates a relationship between derivatives of variables x and y, while the right side of the equation suggests a function of x alone. Without a specific method provided, one can assume standard techniques for solving differential equations could be used here, such as separation of variables, integrating factor, or perturbation techniques for an oscillating solution Yk (x) = Ak cos kx + Bk sin kx, depending on whether it's an ordinary differential equation or a perturbed equation as hinted in the provided information
To approach a general solution, we often start by rearranging terms to either separate variables or identify an integrating factor. If the equation was intended to be dy/dx + 9y = 1/ cos3x −15x+7+e^x, it would become a first-order linear differential equation, solvable via the integrating factor method. However, the general solution will vary significantly based on the correct form of the equation.