Heat Equation Mixed 6D2Dcc

1. **State the problem:** We want to find all separable solutions to the heat equation $$u_t = u_{xx}$$ on the interval $$0 \leq x \leq \pi$$ with boundary conditions $$u(t,0) = 0$$ and $$u_x(t,0) = 0$$. 2. **Use separation of variables:** Assume a solution of the form $$u(t,x) = T(t)X(x)$$. 3. **Substitute into the PDE:** $$u_t = T'(t)X(x), \quad u_{xx} = T(t)X''(x)$$ So the PDE becomes $$T'(t)X(x) = T(t)X''(x)$$ Divide both sides by $$T(t)X(x)$$ (assuming nonzero): $$\frac{T'(t)}{T(t)} = \frac{X''(x)}{X(x)} = -\lambda$$ where $$\lambda$$ is a separation constant. 4. **Form two ODEs:** $$T'(t) + \lambda T(t) = 0$$ $$X''(x) + \lambda X(x) = 0$$ 5. **Apply boundary conditions:** Given $$u(t,0) = 0 \Rightarrow T(t)X(0) = 0$$ for all $$t$$, so $$X(0) = 0$$. Also, $$u_x(t,0) = 0 \Rightarrow T(t)X'(0) = 0$$ for all $$t$$, so $$X'(0) = 0$$. 6. **Analyze boundary conditions on $$X(x)$$:** We have $$X(0) = 0$$ and $$X'(0) = 0$$. 7. **Solve the spatial ODE:** - For $$\lambda = 0$$: $$X'' = 0 \Rightarrow X(x) = A + Bx$$ Apply $$X(0) = 0 \Rightarrow A = 0$$ Apply $$X'(0) = B = 0$$ So $$X(x) = 0$$ trivial solution. - For $$\lambda = \mu^2 > 0$$: $$X'' + \mu^2 X = 0 \Rightarrow X(x) = C \cos(\mu x) + D \sin(\mu x)$$ Apply $$X(0) = C = 0$$ Apply $$X'(x) = D \mu \cos(\mu x)$$, so $$X'(0) = D \mu = 0 \Rightarrow D = 0$$ Thus $$X(x) = 0$$ trivial again. - For $$\lambda = -\alpha^2 < 0$$: $$X'' - \alpha^2 X = 0 \Rightarrow X(x) = E e^{\alpha x} + F e^{-\alpha x}$$ Apply $$X(0) = E + F = 0 \Rightarrow F = -E$$ Apply $$X'(x) = \alpha E e^{\alpha x} - \alpha F e^{-\alpha x}$$ At $$x=0$$: $$X'(0) = \alpha E - \alpha F = \alpha (E - F) = \alpha (E - (-E)) = 2 \alpha E = 0 \Rightarrow E = 0$$ So $$X(x) = 0$$ trivial. 8. **Conclusion:** All boundary conditions force the trivial solution $$X(x) = 0$$, so no nontrivial separable solutions exist. **Final answer:** The only separable solution satisfying $$u(t,0) = 0$$ and $$u_x(t,0) = 0$$ is the trivial solution $$u(t,x) = 0$$.