Solve the differential equation du/dt = (5 + t⁴) / (ut² + u⁴t²)

Solution:

This differential equation can be solved by variable separable method.

A differential equation is said to be in variable separable form if it can be represented in the form, 

f(x)dx = g(y)dy

Given, du/dt = (5 + t⁴)/ (ut² + u⁴t²)

du/dt = (5 + t⁴)/ t²(u + u⁴)

Sorting like terms,

(u + u⁴) du = (5 + t⁴)/ t² .dt

(u + u⁴) du = (5t - 2 + t²) dt

Integrating on both sides,

∫(u + u⁴) du = ∫(5t - 2 + t²) dt

u² / 2 + u⁵ / 5 = 5t²- 2t+ t³/ 3 + C

Solve the differential equation du/dt = (5 + t⁴) / (ut² + u⁴t²)

Summary:

The solution for the differential equation du/dt = (5 + t⁴) / (ut² + u⁴t²) is u² / 2 + u⁵ / 5 = 5t² -2t + t³ / 3 + C.