Final answer:
To solve the differential equation by separation of variables, eʸ and e⁴ terms are separated, integrals are taken with respect to their respective variables, and antiderivatives are found.
Step-by-step explanation:
To solve the given differential equation by separation of variables, we start with the equation:
(eʸ + 1)²e⁻ʸ dx + (e⁴ + 1)⁵e⁴ dy = 0.
We separate the variables by moving all terms involving x to one side and all terms involving y to the other:
- Multiply through by eʸ to eliminate the e⁻ʸ term from the first part of the equation, yielding:
(eʸ + 1)² dx = - (e⁴ + 1)⁵e⁴ dy. - Integrate both sides of the equation:
- For the left side, integrate with respect to x: ∫ (eʸ + 1)² dx
- For the right side, integrate with respect to y: ∫ - (e⁴ + 1)⁵e⁴ dy
Find the antiderivatives and add the constant of integration C.Solve for y as a function of x if required.
Performing the actual integration requires additional steps and may result in a complex expression, including exponential and polynomial terms.