Question

In this problem you will use undetermined coefficients to solve the nonhomogeneous equation y'' - 4y' + 4y = 6e^{2t} - 18te^{2t} + 8t + 4 with initial values y(0) = 5 and y'(0) = 8. A. Write the characteristic equation for the associated homogeneous equation. (Use r for your variable.) r^2 - 4r + 4 = 0 B. Write the fundamental solutions for the associated homogeneous equation. y1 = c1e^{2t} y2 = c2te^{2t} C. Write the form of the particular solution and its derivatives. (Use A, B, C, etc. for undetermined coefficients.) Y = (At^3 + Bt^2 + Ct + D)e^{2t} Y' = (3At^2 + 2Bt + C)e^{2t} + 2(At^3 + Bt^2 + Ct + D)e^{2t} Y'' = (6At + 2B)e^{2t} + 4(3At^2 + 2Bt + C)e^{2t} + 4(At^3 + Bt^2 + Ct + D)e^{2t} D. Write the general solution. (Use c1 and c2 for c1 and c2). y = c1e^{2t} + c2te^{2t} - 3t^3e^{2t} + 3t^2e^{2t} + 2t + 3 E. Plug in the initial values and solve for c1 and c2 to find the solution to the initial value problem. y = -2e^{2t} - te^{2t} - 3t^3e^{2t} + 3t^2e^{2t} + 2t + 3 Hint: No fractions are required in the solution or answer to this problem.

          In this problem you will use undetermined coefficients to solve the nonhomogeneous equation
y'' - 4y' + 4y = 6e^{2t} - 18te^{2t} + 8t + 4
with initial values y(0) = 5 and y'(0) = 8.
A. Write the characteristic equation for the associated homogeneous equation. (Use r for your variable.)
r^2 - 4r + 4 = 0
B. Write the fundamental solutions for the associated homogeneous equation.
y1 = c1e^{2t} y2 = c2te^{2t}
C. Write the form of the particular solution and its derivatives. (Use A, B, C, etc. for undetermined coefficients.)
Y = (At^3 + Bt^2 + Ct + D)e^{2t}
Y' = (3At^2 + 2Bt + C)e^{2t} + 2(At^3 + Bt^2 + Ct + D)e^{2t}
Y'' = (6At + 2B)e^{2t} + 4(3At^2 + 2Bt + C)e^{2t} + 4(At^3 + Bt^2 + Ct + D)e^{2t}
D. Write the general solution. (Use c1 and c2 for c1 and c2).
y = c1e^{2t} + c2te^{2t} - 3t^3e^{2t} + 3t^2e^{2t} + 2t + 3
E. Plug in the initial values and solve for c1 and c2 to find the solution to the initial value problem.
y = -2e^{2t} - te^{2t} - 3t^3e^{2t} + 3t^2e^{2t} + 2t + 3
Hint: No fractions are required in the solution or answer to this problem.

$In this problem you will use undetermined coefficients to solve the nonhomogeneous equation y” - 4y' + 4y = 6e^2t - 18te^2t + 8t + 4 with initial values y(0) = 5 and y'(0) = 8. A. Write the characteristic equation for the associated homogeneous equation. (Use r for your variable.) r^2 - 4r + 4 = 0 B. Write the fundamental solutions for the associated homogeneous equation. y1 = c1e^2t y2 = c2te^2t C. Write the form of the particular solution and its derivatives. (Use A, B, C, etc. for undetermined coefficients.) Y = (At^3 + Bt^2 + Ct + D)e^2t Y' = (3At^2 + 2Bt + C)e^2t + 2(At^3 + Bt^2 + Ct + D)e^2t Y” = (6At + 2B)e^2t + 4(3At^2 + 2Bt + C)e^2t + 4(At^3 + Bt^2 + Ct + D)e^2t D. Write the general solution. (Use c1 and c2 for c1 and c2). y = c1e^2t + c2te^2t - 3t^3e^2t + 3t^2e^2t + 2t + 3 E. Plug in the initial values and solve for c1 and c2 to find the solution to the initial value problem. y = -2e^2t - te^2t - 3t^3e^2t + 3t^2e^2t + 2t + 3 Hint: No fractions are required in the solution or answer to this problem.$

Added by Marcus B.

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