Hello, I'm trying to solve a differential equation using variation of parameters, but I'm not sure what I am doing wrong. The differential equation can also be solved by the method of undetermined coefficients, but the homework wants us to practice variation of parameters.
Any how the question:
Solve y'' - y' = t - e^-t
using variation of parameters
I've solved this using the method of undetermined coefficients and the answer i get is
C1 + C2*e^t - (t^2)/2 - T - (e^-t)/2
when I solve using the variation of parameters the answer I get is similar, can you please let me know what I'm doing wrong.
First I solved for Yh and I got Yh = C1 + C2*e^t
Then I solved for the Wronskian
Wronskian = det [1 e^t]
[0 e^t]
Wronskian = e^t
Then I solved W1 and W2
W1 = [0 e^t]
[t - e^-t e^t]
W1 = 1 - t*e^-t
W2 = [1 0]
[0 t - e^-t]
W2 = t - e^-t
Solve u1 and u2
u1 = integral [1 - t*e^-t]/(e^t) dt
u1 = -e^-t + te^(-2t)/2 + e^(-2*t)/4
u2 = integral [(t- e^-t)/e^t] dt
u2 = e^(-2t)/2 - te^-t - e^-t
solve Yp = u1y1 + u2y2
Yp = -e^-t/2 + te^(-2t)/2 + e^(-2*t)/4 -t - 1
combine Yh and Yp
Y(t) = C1 + C2e^t - e^-t/2 + te^(-2t)/2 + e^(-2t)/4 - t - 1
This is close to the answer i got from method of undetermined coefficients which was
Y(t) = C1 + C2e^t - e^-t/2 + te^(-2t)/2 + e^(-2t)/4 - (t^2)/2 - t
I seem to be off on the last two terms.. Any help would be appreciated on this question.
Thank you!
edit.. I see where I went wrong, I flipped a negative sign
W1 should be 1 - t*e^t
instead of W1 = 1 - t*e^-t
Any how the question:
Solve y'' - y' = t - e^-t
using variation of parameters
I've solved this using the method of undetermined coefficients and the answer i get is
C1 + C2*e^t - (t^2)/2 - T - (e^-t)/2
when I solve using the variation of parameters the answer I get is similar, can you please let me know what I'm doing wrong.
First I solved for Yh and I got Yh = C1 + C2*e^t
Then I solved for the Wronskian
Wronskian = det [1 e^t]
[0 e^t]
Wronskian = e^t
Then I solved W1 and W2
W1 = [0 e^t]
[t - e^-t e^t]
W1 = 1 - t*e^-t
W2 = [1 0]
[0 t - e^-t]
W2 = t - e^-t
Solve u1 and u2
u1 = integral [1 - t*e^-t]/(e^t) dt
u1 = -e^-t + te^(-2t)/2 + e^(-2*t)/4
u2 = integral [(t- e^-t)/e^t] dt
u2 = e^(-2t)/2 - te^-t - e^-t
solve Yp = u1y1 + u2y2
Yp = -e^-t/2 + te^(-2t)/2 + e^(-2*t)/4 -t - 1
combine Yh and Yp
Y(t) = C1 + C2e^t - e^-t/2 + te^(-2t)/2 + e^(-2t)/4 - t - 1
This is close to the answer i got from method of undetermined coefficients which was
Y(t) = C1 + C2e^t - e^-t/2 + te^(-2t)/2 + e^(-2t)/4 - (t^2)/2 - t
I seem to be off on the last two terms.. Any help would be appreciated on this question.
Thank you!
edit.. I see where I went wrong, I flipped a negative sign
W1 should be 1 - t*e^t
instead of W1 = 1 - t*e^-t
Last edited: