Solve the differential equation. du/dt = 9 + 9u + t + tu.

Solution:

Given differential equation, du/dt = 9 + 9u + t + tu

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

du/dt = 9(1 + u) + t(1 + u)

du/dt = (9 + t) (1 + u)

Sorting the like terms,

du/(1 + u) = (9 + t) dt

Integrating both the sides

∫[1/(1+u)] du = ∫ (9 + t) dt

Log |(1 + u)| = [9t + t² / 2] + C

|1+u| = e^{(9t + t² / 2 + C)}

|1+u| = ke^{(9t + t² / 2)} (Where k = e^C)

u= ±ke^{(9t + t² / 2)} -1

u= Ae^{(9t + t² / 2)} -1 (where A = ±k = an arbitrary constant)

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Solve the differential equation. du/dt = 9 + 9u + t + tu.

Summary:

The solution for the given differential equation du/dt = 9 + 9u + t + tu is Log (1 + u) = u= Ae^{(9t + t² / 2)} -1