Question

(25 points) In this problem you will use undetermined coefficients to solve the nonhomogeneous equation y'' - y' - 6y = 20te^{3t} + 14e^{3t} - (6t + 13) with initial values y(0) = -2 and y'(0) = -4. A. Write the characteristic equation for the associated homogeneous equation. (Use r for your variable.) r^2-r-6=0 B. Write the fundamental solutions for the associated homogeneous equation. y_1 = e^(3t) y_2 = e^(-2t) C. Write the form of the particular solution and its derivatives. (Use A, B, C, etc. for undetermined coefficients. Y = At+B+(Ct+D)te^(3t) Y' = Y'' = D. Write the general solution. (Use c1 and c2 for c_1 and c_2). y = c1e^(3t)+c2e^(-2t)+2e^(3t)t^2+t+2 E. Plug in the initial values and solve for c_1 and c_2 to find the solution to the initial value problem. y = -3e^(3t)-e^(-2t)+2e^(3t)t^2+t+2

          (25 points) In this problem you will use undetermined coefficients to solve the nonhomogeneous equation
y'' - y' - 6y = 20te^{3t} + 14e^{3t} - (6t + 13)
with initial values y(0) = -2 and y'(0) = -4.
A. Write the characteristic equation for the associated homogeneous equation. (Use r for your variable.)
r^2-r-6=0
B. Write the fundamental solutions for the associated homogeneous equation.
y_1 = e^(3t) y_2 = e^(-2t)
C. Write the form of the particular solution and its derivatives. (Use A, B, C, etc. for undetermined coefficients.
Y = At+B+(Ct+D)te^(3t)
Y' =
Y'' =
D. Write the general solution. (Use c1 and c2 for c_1 and c_2).
y = c1e^(3t)+c2e^(-2t)+2e^(3t)t^2+t+2
E. Plug in the initial values and solve for c_1 and c_2 to find the solution to the initial value problem.
y = -3e^(3t)-e^(-2t)+2e^(3t)t^2+t+2

(25 points) In this problem you will use undetermined coefficients to solve the nonhomogeneous equation
y” - y' - 6y = 20te^3t + 14e^3t - (6t + 13)
with initial values y(0) = -2 and y'(0) = -4.
A. Write the characteristic equation for the associated homogeneous equation. (Use r for your variable.)
r^2-r-6=0
B. Write the fundamental solutions for the associated homogeneous equation.
y1 = e^(3t) y2 = e^(-2t)
C. Write the form of the particular solution and its derivatives. (Use A, B, C, etc. for undetermined coefficients.
Y = At+B+(Ct+D)te^(3t)
Y' =
Y” =
D. Write the general solution. (Use c1 and c2 for c1 and c2).
y = c1e^(3t)+c2e^(-2t)+2e^(3t)t^2+t+2
E. Plug in the initial values and solve for c1 and c2 to find the solution to the initial value problem.
y = -3e^(3t)-e^(-2t)+2e^(3t)t^2+t+2

Added by Fevzi Ö.

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