To solve the differential equation using separation of variables, the equation is rewritten to separate the variables and then integrated to find the general solution.
To solve the given differential equation by separation of variables, we first need to rewrite the equation into a separable form. Starting with eⁱy dy/dx = e⁻¹ + e⁻⁵ⁱ⁻¹, we can divide both sides by eⁱy, which gives us:
dy/dx = (e⁻¹ + e⁻⁵ⁱ⁻¹) / eⁱy
Now, we can separate variables by moving all terms with y to one side of the equation and all terms with x to the other side:
dy / (e⁻¹ + e⁻⁵ⁱ⁻¹) = dx / eⁱy
To complete the process, both sides of the equation can be integrated to find the general solution to the differential equation. Without additional initial conditions, we will have an arbitrary constant in our solution.
Our final step would be to solve the resulting integral, which may require substitution and knowledge of integral calculus to evaluate.