User:Egm4313.s12.team11.Suh/R6.7

Report 6, Problem 6.7
Solved by Solved by Daniel Suh

Problem Statement
Find the separated Ordinary Differential Equations for the heat equation.

$$\frac{\partial u}{\partial t}=\kappa \frac{\partial^2 u}{\partial x^2} \cdots (1)\!$$

$$u(x,t) = F(x)\cdot G(t)\!$$

Solution
$$\frac{\partial u}{\partial t} = F(x)\cdot \dot{G}(t) \cdots (2)\!$$

$$\frac{\partial^2 u}{\partial x^2} = F''(x)\cdot G(t) \cdots (3)\!$$

Combine $$(2)\!$$ and $$(3)\!$$ into $$(1)\!$$ 

$$F(x)\cdot \dot{G}(t) = \kappa F''(x)\cdot G(t)\!$$

$$\frac{\dot{G}(t)}{\kappa G(t)} = \frac{F''(x)}{F(x)} = c\!$$ (constant)

Separate

$$F''(x)-cF(x) = 0\!$$ $$G'(t)-\kappa cG(t) = 0\!$$