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
To find the inverse Laplace transform of the given function, we first perform partial fraction decomposition. The function is: F(s) = (1)/(s^2(s+1)(s^2+9)) Step 1: Partial Fraction Decomposition We set up the partial fraction decomposition as follows: (1)/(s^2(s+1)(s^2+9)) = (A)/(s) + (B)/(s^2) + (C)/(s+1) + (Ds+E)/(s^2+9) Multiply both sides by s^2(s+1)(s^2+9): 1 = A s(s+1)(s^2+9) + B (s+1)(s^2+9) + C s^2(s^2+9) + (Ds+E) s^2(s+1) To find the coefficients, we can substitute specific values for s: Set s=0: 1 = A(0) + B(0+1)(0^2+9) + C(0) + (D(0)+E)(0) 1 = B(1)(9) 9B = 1 B = (1)/(9) Set s=-1: 1 = A(-1)(-1+1)((-1)^2+9) + B(-1+1)((-1)^2+9) + C(-1)^2((-1)^2+9) + (D(-1)+E)(-1)^2(-1+1) 1 = A(0) + B(0) + C(1)(1+9) + (D(-1)+E)(0) 1 = 10C C = (1)/(10) Now, expand the equation and equate coefficients of powers of s: 1 = A(s^4+s^3+9s^2+9s) + B(s^3+s^2+9s+9) + C(s^4+9s^2) + (Ds+E)(s^3+s^2) 1 = A(s^4+s^3+9s^2+9s) + B(s^3+s^2+9s+9) + C(s^4+9s^2) + (Ds^4+Ds^3+Es^3+Es^2) 1 = (A+C+D)s^4 + (A+B+D+E)s^3 + (9A+B+9C+E)s^2 + (9A+9B)s + 9B Equating coefficients: Constant term: 1 = 9B. This confirms B = (1)/(9). Coefficient of s: 0 = 9A+9B Substitute B = (1)/(9): 0 = 9A + 9((1)/(9)) 0 = 9A + 1 A = -(1)/(9) Coefficient of s^4: 0 = A+C+D Substitute A = -(1)/(9) and C = (1)/(10): 0 = -(1)/(9) + (1)/(10) + D D = (1)/(9) - (1)/(10) = (10-9)/(90) = (1)/(90)