Final answer:
This question involves solving a separable differential equation with an initial condition in the field of mathematics. The equation requires separation of variables and integration, applying the initial condition to find the constant of integration.
Step-by-step explanation:
The student's question involves solving a separable differential equation 5x-8y√x²+1 dy/dx=0 with the initial condition y(0)=8. This is a mathematical problem, specifically in the realm of differential equations, which are equations that involve derivatives of functions and are solved to determine the functions themselves.
Firstly, we can separate the variables by dividing both sides by √x²+1 and then by 5x-8y, assuming that 5x-8y is not equal to zero. After separation, we will integrate both sides with respect to their respective variables, apply the initial condition to find the constant of integration, and solve for y in terms of x if possible.
However, without further simplification or context, it is impossible to provide a precise solution to this equation. This is because the equation is directly given as equal to zero, which, assuming the separable form is 5x dy - 8y(√x²+1)dx = 0, suggests that either dy or dx could be zero, depending on the context or domain of the problem. As such, specific methods from calculus would be employed for different scenarios, which have not been outlined in the question provided.