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The problem shown in the image asks for the equilibrium temperature distribution in a heat problem described by a partial differential equation (PDE), boundary conditions (BC), and initial conditions (IC).

Here’s a breakdown of the problem:

### 1. **PDE:**
\[
2 \frac{\partial u}{\partial t} = K_0 \frac{\partial^2 u}{\partial x^2} + K_0 x, \quad 0 < x < 1, \, t > 0
\]
This equation describes the temperature evolution \( u(x,t) \) over time and space. The term \( K_0 x \) represents a source term that depends on \( x \), while \( \frac{\partial^2 u}{\partial x^2} \) is the diffusion term.

### 2. **Boundary Conditions (BC):**
\[
u(0, t) = T, \quad \frac{\partial u}{\partial x}(1,t) + u(1,t) = 0, \quad t > 0
\]
- At \( x = 0 \), the temperature is fixed at \( T \).
- At \( x = 1 \), we have a mixed boundary condition: the derivative of \( u \) with respect to \( x \) plus the temperature at \( x = 1 \) equals zero.

### 3. **Initial Condition (IC):**
\[
u(x,0) = \sin(x), \quad 0 < x < 1
\]
This specifies the temperature distribution at time \( t = 0 \).

### **Equilibrium Solution:**
At equilibrium, the temperature distribution no longer changes with time, meaning the time derivative \( \frac{\partial u}{\partial t} \) is zero. This simplifies the PDE to a steady-state form:

\[
0 = K_0 \frac{\partial^2 u}{\partial x^2} + K_0 x
\]
or
\[
\frac{\partial^2 u}{\partial x^2} = -x
\]
This is a second-order ordinary differential equation (ODE) for \( u(x) \).

### **Solution of the ODE:**
To solve this ODE:
\[
\frac{d^2 u}{dx^2} = -x
\]
Integrating twice:
1. First integration gives:
   \[
   \frac{du}{dx} = -\frac{x^2}{2} + C_1
   \]
2. Second integration gives:
   \[
   u(x) = -\frac{x^3}{6} + C_1 x + C_2
   \]

### **Applying the Boundary Conditions:**
- From the boundary condition \( u(0,t) = T \), we get:
  \[
  u(0) = T \implies C_2 = T
  \]
- From the boundary condition \( \frac{\partial u}{\partial x}(1,t) + u(1,t) = 0 \), we use:
  \[
  \frac{du}{dx}(1) + u(1) = 0
  \]
  Substituting \( u(x) = -\frac{x^3}{6} + C_1 x + T \):
  \[
  \left( -\frac{1^2}{2} + C_1 \right) + \left( -\frac{1^3}{6} + C_1 \cdot 1 + T \right) = 0
  \]
  Simplifying this gives:
  \[
  \left( -\frac{1}{2} + C_1 \right) + \left( -\frac{1}{6

Determine the equilibrium temperature distribution for a heat problem described by the given PDE, boundary conditions, and initial conditions.

Explore the determination of the equilibrium temperature distribution for a heat problem involving a partial differential equation with a source term. This advanced mathematical problem involves solving a PDE, applying mixed boundary conditions, and computing the steady state solution.

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