This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Here are the solutions to the Laplace Transform problems: 1. Find L^-1[(1)/(s^3)] Step 1: Use the inverse Laplace transform formula L^-1[(n!)/(s^n+1)] = t^n. Here, n+1=3, so n=2. We need 2! in the numerator. L^-1[(1)/(s^3)] = L^-1[(1)/(2!) · (2!)/(s^3)] Step 2: Apply the formula. L^-1[(1)/(s^3)] = (1)/(2) L^-1[(2!)/(s^3)] = (1)/(2) t^2 The final answer is (1)/(2) t^2. 2. Find L^-1[(1)/(s^2(s^2+1)(s^2+9))] Step 1: Use partial fraction decomposition. Let X = s^2. (1)/(X(X+1)(X+9)) = (A)/(X) + (B)/(X+1) + (C)/(X+9) Step 2: Solve for A, B, C. For A, set X=0: A = (1)/((0+1)(0+9)) = (1)/(9). For B, set X=-1: $B = (1)/((-1)(-1+9)) = (1)/(-8) = -