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To solve this problem, we need to determine the function \( v(x) \) such that the substitution \( T(x, t) = v(x) + \phi(x, t) \) transforms the given inhomogeneous partial differential equation into a homogeneous one.

### Step-by-Step Solution:

The given inhomogeneous PDE is:

\[
\frac{\partial}{\partial x} \left( \frac{1}{3 + 2x^2} \frac{\partial T}{\partial x} \right) = 4 + \frac{\partial T}{\partial t} + \frac{\partial^2 T}{\partial t^2}.
\]

Substituting \( T(x, t) = v(x) + \phi(x, t) \), we want to find \( v(x) \) such that the resulting PDE in \(\phi(x, t)\) is homogeneous (i.e., without any external terms like constants or functions of \(x\) only).

### Derivatives of \( T \):

1. First, compute the partial derivatives of \( T(x, t) \):
   - Since \( T(x, t) = v(x) + \phi(x, t) \):
     \[
     \frac{\partial T}{\partial x} = \frac{\partial v(x)}{\partial x} + \frac{\partial \phi(x, t)}{\partial x},
     \]
     \[
     \frac{\partial T}{\partial t} = \frac{\partial \phi(x, t)}{\partial t},
     \]
     \[
     \frac{\partial^2 T}{\partial t^2} = \frac{\partial^2 \phi(x, t)}{\partial t^2}.
     \]

2. Substitute these into the given PDE:

   \[
   \frac{\partial}{\partial x} \left( \frac{1}{3 + 2x^2} \left( \frac{\partial v(x)}{\partial x} + \frac{\partial \phi(x, t)}{\partial x} \right) \right) = 4 + \frac{\partial \phi(x, t)}{\partial t} + \frac{\partial^2 \phi(x, t)}{\partial t^2}.
   \]

3. Separate the terms involving only \( v(x) \):

   The inhomogeneous term \( 4 \) must come from the part involving \( v(x) \), which means we set up the equation for \( v(x) \) to satisfy:

   \[
   \frac{\partial}{\partial x} \left( \frac{1}{3 + 2x^2} \frac{\partial v(x)}{\partial x} \right) = 4.
   \]

### Solving for \( v(x) \):

1. Integrate both sides with respect to \(x\):

   \[
   \frac{1}{3 + 2x^2} \frac{\partial v(x)}

Consider the inhomogeneous partial differential equation in T(x,t) given by: \[ \frac{\partial}{\partial x} \left( \frac{1}{3 + 2x^2} \frac{\partial T}{\partial x} \right) = 4 + \frac{\partial T}{\partial t} + \frac{\partial^2 T}{\partial t^2}. \] Choose v(x) such a way that the substitution T(x, t) = v(x) + \phi(x,t) gives rise to a homogeneous partial differential equation in \phi(x,t).

Learn how to solve the inhomogeneous partial differential equation involving T(x, t) and find the appropriate function v(x) to transform the equation into a homogeneous PDE in \phi(x,t). This problem applies separation of variables and inhomogeneous PDE techniques, suitable for undergraduate mathematics students.

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