I am working on a review for an exam, and can't seem to arrive at the same answer as the textbook. The instructions are to find a general solution x(t) and y(t) to the system.
2x' - y' = y + 3x +e^t
3y' -4x' = y -15x + e^(-t)
To start, I rewrote the equations in operator form
(2D - 3)x + (-D-1)y = e^t
(-4D + 15)x + (3D - 1)y = e^-t)
I followed some steps outlined in the textbook, where
L1[x] + L2[y] = f1
L3[x] + L4[y] = f2
With L1 being (2D - 3), L2 being (-D - 1), L3 (-4D + 15), L4 (3D - 1)
So, (L1*L4 - L2*L3)[x] = g1, with g1 being L4[f1] - L2[f2]
And (L1*L4 - L2*L3)[y] = g2, with g2 being L1[f2] - L3[f1]
I did the math for x(t), and wound up with the correct answer for that part, C1 cos(3t) + C2 sin(3t) + e^t/10
I repeated the process for y(t), and my answer was C1 cos(3t) + C2 sin(3t) - 1/2 e^(-t) - 11/10 e^t
This doesn't match the answer in the book, which is y(t) = 3/2 (C1 - C2) cos(3t) - 3/2 (C1 - C2) sin(3t) - 11/20 e^t - 1/4 e^(-t)
I'm missing something, and don't know what I'm doing wrong for y(t).
It seems to me that using the L1 etc. method of elimination should lead to the homogeneous portion of both x(t) and y(t) being the same, with the particular portion of y(t) being different because g2 ends up with two terms.
I'd appreciate if someone could help me out with this. I did really well in Calc I, II, and III, but this course is giving me a hard time.
2x' - y' = y + 3x +e^t
3y' -4x' = y -15x + e^(-t)
To start, I rewrote the equations in operator form
(2D - 3)x + (-D-1)y = e^t
(-4D + 15)x + (3D - 1)y = e^-t)
I followed some steps outlined in the textbook, where
L1[x] + L2[y] = f1
L3[x] + L4[y] = f2
With L1 being (2D - 3), L2 being (-D - 1), L3 (-4D + 15), L4 (3D - 1)
So, (L1*L4 - L2*L3)[x] = g1, with g1 being L4[f1] - L2[f2]
And (L1*L4 - L2*L3)[y] = g2, with g2 being L1[f2] - L3[f1]
I did the math for x(t), and wound up with the correct answer for that part, C1 cos(3t) + C2 sin(3t) + e^t/10
I repeated the process for y(t), and my answer was C1 cos(3t) + C2 sin(3t) - 1/2 e^(-t) - 11/10 e^t
This doesn't match the answer in the book, which is y(t) = 3/2 (C1 - C2) cos(3t) - 3/2 (C1 - C2) sin(3t) - 11/20 e^t - 1/4 e^(-t)
I'm missing something, and don't know what I'm doing wrong for y(t).
It seems to me that using the L1 etc. method of elimination should lead to the homogeneous portion of both x(t) and y(t) being the same, with the particular portion of y(t) being different because g2 ends up with two terms.
I'd appreciate if someone could help me out with this. I did really well in Calc I, II, and III, but this course is giving me a hard time.
