Math Problem Statement
2 Find the function u(x, t), dened for 0 ≤ x ≤ π and t ≥ 0, which satises the following conditions: ∂u ∂t = ∂ 2 u ∂x2 , u(0, t) = u(π, t) = 0, u(x, 0) = sin(x) + 3sin(2x) − 5sin(3x).
Solution
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Math Problem Analysis
Mathematical Concepts
Partial Differential Equations
Heat Equation
Separation of Variables
Formulas
∂u/∂t = ∂²u/∂x² (Heat equation)
u(x, t) = X(x)T(t) (Separation of Variables)
General solution: u(x, t) = ΣA_n sin(nx) e^(-n²t)
Theorems
Separation of Variables
Boundary Value Problem Solutions
Suitable Grade Level
Undergraduate Level (Math or Physics)
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