Answer to Question #222718 in Differential Equations for liam

Question

y''-3y'+2y=2x²+e^x+2xe^x+4e^3x

Answer 1

"\\displaystyle\n\\textsf{The auxiliary equation is}\\\\\nm^2 - 3m + 2 = 0\\\\\nm = 1, 2\\\\\n\n\\implies y = C_1e^x + C_2e^{2x}\\\\\n\ny''-3y'+2y=2x^2+e^x+2xe^x+4e^(3x)\\\\\n\ny = Ax^2 + Bx + C + Dxe^x + Ex^2e^x + Fe^{3x}\\\\\ny' = 2Ax + B + De^x + Dxe^x + 2Exe^x + Ex^2e^x + 3Fe^{3x}\\\\\n\\begin{aligned}\ny'' &= 2A + 2De^x + Dxe^x + 2Ee^x + 2Exe^x \\\\&+ Ex^2e^x + 2Exe^x + 9Fe^{3x}\n\\\\&= 2A + (2D + 2E)e^x + (D + 4E)xe^x + Ex^2e^x + 9Fe^{3x}\n\\end{aligned}\\\\\n\ny' = 2Ax + B + De^x + (D + 2E)xe^x + Ex^2e^x + 3Fe^{3x}\\\\\ny = Ax^2 + Bx + C + Dxe^x + Ex^2e^x + Fe^{3x}\\\\\n\n\\begin{aligned}\ny" - 3y' + 2y &= (2A - 3B + 2C) + (2E - D)e^x - 2Exe^x \n\\\\&+ 2Fe^{3x} + (2B - 6A)x + 2Ax^2 \n\\\\&= 2x^2+e^x+2xe^x+4e^(3x)\n\\end{aligned}\\\\\n\n2A = 2, A = 1\\\\\n\n2B - 6A = 0, B = 3\\\\\n\n2F = 4, F = 2\\\\\n\n-2E = 2, E = -1\\\\\n\n2E - D = 1, D = -3\\\\\n\n2A - 3B + 2C = 0\\\\\n\n2 - 9 + 2C = 0, C = 7\/2\\\\\n\ny = x^2 + 3x + 7\/2 - 3xe^x - x^2e^x + 2e^{3x}\\\\\n\n\\implies y = C_1e^x + C_2e^{2x} + x^2 + 3x \\\\+ 7\/2 - 3xe^x - x^2e^x + 2e^{3x}"