Final answer:
To solve the differential equation dy/dx = 8xy/(ln y)² is y = 4x² + ln e
Step-by-step explanation:
To find the solution to the differential equation dy/dx = 8xy/(ln y)² that passes through the point (0, e), we need to separate the variables and integrate both sides. Starting with the given differential equation, we can rearrange it as (ln y)² dy = 8xy dx. Integrating both sides gives us ∫(ln y)² dy = ∫8xy dx.
Integrating the left side can be done using u-substitution, while integrating the right side involves manipulating the integrand and applying the power rule. Once both integrals are solved, we can use the initial conditions (0, e) to find a particular solution.
After solving the integrals and applying the initial condition, we can express the solution as ln y = 4x² + ln e.