Final answer:
To solve the given differential equation using separation of variables, one must first separate the x and y terms and then integrate both sides. The process may require specific integration techniques such as partial fractions or substitutions. The final step is to apply initial conditions if available or provide the general solution.
Step-by-step explanation:
To solve the differential equation dy/dx = (2y + 5)/(6x + 7)² by separation of variables, we first begin by separating the variables y and x so that each variable is on one side of the equation. We achieve this by multiplying both sides of the equation by dx and dividing by (2y + 5)², which gives us (1/(2y + 5)²) dy = (1/(6x + 7)²) dx. Next, we integrate both sides of the equation with respect to their respective variables.
For instance, ∫y (1/(2y + 5)²) dy = ∫x (1/(6x + 7)²) dx. Solving these integrals may involve a partial fraction decomposition or a substitution method. Once the integrations are complete, we find the constants of integration, and we can then write down the general solution to the differential equation.
Finally, we can apply any initial conditions given in the problem to find the particular solution. If there are no initial conditions provided, we simply report the general solution.