We consider the non-homogeneous initial value problem z?y" – 8ry' + 20y = -20x°, y(1) = -2, y'(1) = -5 By looking for solutions in the form y = x" in the homogeneous Euler-Cauchy problem Ax?y" + Bxy' + Cy = 0, we obtain a auxiliary equation Ar? + (B – A)r + C = 0 which is the analog of the auxiliary equation in the constant coefficient case. (1) For this problem find the auxiliary equation: = 0 (2) Find the roots of the auxiliary equation: (enter your results as a comma separated list) (3) Find a fundamental set of solutions Y1, 42: (enter your results as a comma separated list ). Then the complementary solution is yYc = C1Y1 + c2Y2. (4) For the non-homogeneous problem we need to apply the variation of parameters formula to find a particular solution . Carrying this out you obtain Yp Recall that the general solution is y = Yc + Yp. Find the unique solution satisfying y(1) =-2, y'(1) =-5 y =
We consider the non-homogeneous initial value problem z?y" – 8ry' + 20y = -20x°, y(1) = -2, y'(1) = -5 By looking for solutions in the form y = x" in the homogeneous Euler-Cauchy problem Ax?y" + Bxy' + Cy = 0, we obtain a auxiliary equation Ar? + (B – A)r + C = 0 which is the analog of the auxiliary equation in the constant coefficient case. (1) For this problem find the auxiliary equation: = 0 (2) Find the roots of the auxiliary equation: (enter your results as a comma separated list) (3) Find a fundamental set of solutions Y1, 42: (enter your results as a comma separated list ). Then the complementary solution is yYc = C1Y1 + c2Y2. (4) For the non-homogeneous problem we need to apply the variation of parameters formula to find a particular solution . Carrying this out you obtain Yp Recall that the general solution is y = Yc + Yp. Find the unique solution satisfying y(1) =-2, y'(1) =-5 y =
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:We consider the non-homogeneous initial value problem z?y" – 8xy' + 20y = -20a°, y(1) = -2, y'(1) = -5
%3D
By looking for solutions in the form y = x" in the homogeneous Euler-Cauchy problem Ax?y"+ Bxy' + Cy= 0, we obtain a
auxiliary equation Ar“ + (B– A)r + C = 0 which is the analog of the auxiliary equation in the constant coefficient case.
%3D
(1) For this problem find the auxiliary equation:
0 =
(2) Find the roots of the auxiliary equation:
(enter your results as a comma separated list )
(3) Find a fundamental set of solutions Y1, Y2:
(enter your results as a comma separated list ). Then the
complementary solution is Yc = C1Y1 + C2Y2-
(4) For the non-homogeneous problem we need to apply the variation of parameters formula to find a particular solution . Carrying
this out you obtain
Yp
Recall that the general solution is y = y. + Yp: Find the unique solution satisfying y(1) = -2, y'(1) = -5
y =
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