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