Math Problem Statement
Consider the IVP d 2 y dt 2 − 6 dy dt + 15y = 2 sin(3t) y(0) = −1, y ′ (0) = −4. (a) Solve the IVP using Laplace Transforms. (b) Express the IVP as a vector-matrix transformation. Apply Laplace Transforms to solve the resultant system of first order ODEs. This offers an alternative approach to the method of matrix diagonalisation from Topic 1.
Solution
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Math Problem Analysis
Mathematical Concepts
Laplace Transforms
Differential Equations
Vector-Matrix Transformation
First-Order Systems
Formulas
Laplace transform of derivatives: L{y'(t)} = sY(s) - y(0), L{y''(t)} = s²Y(s) - sy(0) - y'(0)
Laplace transform of sin(3t): L{sin(3t)} = 3/(s² + 9)
Matrix form of first-order system: d/dt [y1; y2] = [0 1; -15 6] [y1; y2] + [0; 2 sin(3t)]
Theorems
Laplace Transform Theorem
Matrix Theory for First-Order Systems
Suitable Grade Level
Undergraduate level (Advanced Calculus/ODE)
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