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Simplifying (D + 1)(D2 + 5D + 6) * y(t) = (5D + 9) * f(t) Reorder the terms: (1 + D)(D2 + 5D + 6) * y(t) = (5D + 9) * f(t) Reorder the terms: (1 + D)(6 + 5D + D2) * y(t) = (5D + 9) * f(t) Reorder the terms for easier multiplication: y * t(1 + D)(6 + 5D + D2) = (5D + 9) * f(t) Multiply y * t ty(1 + D)(6 + 5D + D2) = (5D + 9) * f(t) Multiply (1 + D) * (6 + 5D + D2) ty(1(6 + 5D + D2) + D(6 + 5D + D2)) = (5D + 9) * f(t) ty((6 * 1 + 5D * 1 + D2 * 1) + D(6 + 5D + D2)) = (5D + 9) * f(t) ty((6 + 5D + 1D2) + D(6 + 5D + D2)) = (5D + 9) * f(t) ty(6 + 5D + 1D2 + (6 * D + 5D * D + D2 * D)) = (5D + 9) * f(t) ty(6 + 5D + 1D2 + (6D + 5D2 + D3)) = (5D + 9) * f(t) Reorder the terms: ty(6 + 5D + 6D + 1D2 + 5D2 + D3) = (5D + 9) * f(t) Combine like terms: 5D + 6D = 11D ty(6 + 11D + 1D2 + 5D2 + D3) = (5D + 9) * f(t) Combine like terms: 1D2 + 5D2 = 6D2 ty(6 + 11D + 6D2 + D3) = (5D + 9) * f(t) (6 * ty + 11D * ty + 6D2 * ty + D3 * ty) = (5D + 9) * f(t) (6ty + 11tyD + 6tyD2 + tyD3) = (5D + 9) * f(t) Reorder the terms: 6ty + 11tyD + 6tyD2 + tyD3 = (9 + 5D) * f(t) Reorder the terms for easier multiplication: 6ty + 11tyD + 6tyD2 + tyD3 = f * t(9 + 5D) Multiply f * t 6ty + 11tyD + 6tyD2 + tyD3 = ft(9 + 5D) 6ty + 11tyD + 6tyD2 + tyD3 = (9 * ft + 5D * ft) 6ty + 11tyD + 6tyD2 + tyD3 = (9ft + 5ftD) Solving 6ty + 11tyD + 6tyD2 + tyD3 = 9ft + 5ftD Solving for variable 't'. Move all terms containing t to the left, all other terms to the right. Add '-9ft' to each side of the equation. 6ty + 11tyD + 6tyD2 + -9ft + tyD3 = 9ft + -9ft + 5ftD Reorder the terms: -9ft + 6ty + 11tyD + 6tyD2 + tyD3 = 9ft + -9ft + 5ftD Combine like terms: 9ft + -9ft = 0 -9ft + 6ty + 11tyD + 6tyD2 + tyD3 = 0 + 5ftD -9ft + 6ty + 11tyD + 6tyD2 + tyD3 = 5ftD Add '-5ftD' to each side of the equation. -9ft + 6ty + 11tyD + 6tyD2 + -5ftD + tyD3 = 5ftD + -5ftD Reorder the terms: -9ft + -5ftD + 6ty + 11tyD + 6tyD2 + tyD3 = 5ftD + -5ftD Combine like terms: 5ftD + -5ftD = 0 -9ft + -5ftD + 6ty + 11tyD + 6tyD2 + tyD3 = 0 Factor out the Greatest Common Factor (GCF), 't'. t(-9f + -5fD + 6y + 11yD + 6yD2 + yD3) = 0Subproblem 1
Set the factor 't' equal to zero and attempt to solve: Simplifying t = 0 Solving t = 0 Move all terms containing t to the left, all other terms to the right. Simplifying t = 0Subproblem 2
Set the factor '(-9f + -5fD + 6y + 11yD + 6yD2 + yD3)' equal to zero and attempt to solve: Simplifying -9f + -5fD + 6y + 11yD + 6yD2 + yD3 = 0 Solving -9f + -5fD + 6y + 11yD + 6yD2 + yD3 = 0 Move all terms containing t to the left, all other terms to the right. Add '9f' to each side of the equation. -9f + -5fD + 6y + 11yD + 6yD2 + 9f + yD3 = 0 + 9f Reorder the terms: -9f + 9f + -5fD + 6y + 11yD + 6yD2 + yD3 = 0 + 9f Combine like terms: -9f + 9f = 0 0 + -5fD + 6y + 11yD + 6yD2 + yD3 = 0 + 9f -5fD + 6y + 11yD + 6yD2 + yD3 = 0 + 9f Remove the zero: -5fD + 6y + 11yD + 6yD2 + yD3 = 9f Add '5fD' to each side of the equation. -5fD + 6y + 11yD + 6yD2 + 5fD + yD3 = 9f + 5fD Reorder the terms: -5fD + 5fD + 6y + 11yD + 6yD2 + yD3 = 9f + 5fD Combine like terms: -5fD + 5fD = 0 0 + 6y + 11yD + 6yD2 + yD3 = 9f + 5fD 6y + 11yD + 6yD2 + yD3 = 9f + 5fD Add '-6y' to each side of the equation. 6y + 11yD + 6yD2 + -6y + yD3 = 9f + 5fD + -6y Reorder the terms: 6y + -6y + 11yD + 6yD2 + yD3 = 9f + 5fD + -6y Combine like terms: 6y + -6y = 0 0 + 11yD + 6yD2 + yD3 = 9f + 5fD + -6y 11yD + 6yD2 + yD3 = 9f + 5fD + -6y Add '-11yD' to each side of the equation. 11yD + 6yD2 + -11yD + yD3 = 9f + 5fD + -6y + -11yD Reorder the terms: 11yD + -11yD + 6yD2 + yD3 = 9f + 5fD + -6y + -11yD Combine like terms: 11yD + -11yD = 0 0 + 6yD2 + yD3 = 9f + 5fD + -6y + -11yD 6yD2 + yD3 = 9f + 5fD + -6y + -11yD Add '-6yD2' to each side of the equation. 6yD2 + -6yD2 + yD3 = 9f + 5fD + -6y + -11yD + -6yD2 Combine like terms: 6yD2 + -6yD2 = 0 0 + yD3 = 9f + 5fD + -6y + -11yD + -6yD2 yD3 = 9f + 5fD + -6y + -11yD + -6yD2 Add '-1yD3' to each side of the equation. yD3 + -1yD3 = 9f + 5fD + -6y + -11yD + -6yD2 + -1yD3 Combine like terms: yD3 + -1yD3 = 0 0 = 9f + 5fD + -6y + -11yD + -6yD2 + -1yD3 Simplifying 0 = 9f + 5fD + -6y + -11yD + -6yD2 + -1yD3 The solution to this equation could not be determined. This subproblem is being ignored because a solution could not be determined.Solution
t = {0}
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