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7) y"+4y+20y-23sint-15cost y(0)-0, y'(0)--1

7) y"+4y+20y-23sint-15cost y(0)-0, y'(0)--1

Advanced Engineering Mathematics
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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7) y"+4y'+20y=23sint-15cost y(0)=0, y'(0)=-1
Transcribed Image Text:7) y"+4y'+20y=23sint-15cost y(0)=0, y'(0)=-1
Methods of finding the particular solution( v,): 1) Undetermined coefficients. We can extend the methods of solving second order non homogenous differential equation with constant coefficients to solve higher order nonhomogeneous differential cquation with constant cocfficients. Example: Solve y-8y"+16y=-18sinx Solution A-87+16-0 = (-4)²-0=r-4=r=+2 Let y,-Acosx+Bsinx , y',--Asinx+Bcosx, y",= -Acosx-Bsinx y",- Asinx-Bcosx , y",- Acosx+Bsinx Acosx+Bsinx+8Acosx+8Bsinx+164cosx+16Bsinx-18sinx 25Acosx+25Bsinx=-18sinx 254-04-0 25B-18->В-18/25 y, =c,e* +cxe* +c;e* 18 sinx 25 a)Variation of parameters In this method, the particular solution y, has the form y,-v,u,+vuz+. +v,l, Where u1, uz, ..., U, are taken from ya=ciuj+c;uz+. +Cntty. To find v, vz, . V, we must solve the following linear equations. For v', v's ., v vu, + vu, +... +vu, =0 vu, + v,i", +... +v = 0 +... +vu a-3 +... +vu," f(x) (n-2) (n-2) vju, vu," =0 (n-I) (e-1)
Transcribed Image Text:Methods of finding the particular solution( v,): 1) Undetermined coefficients. We can extend the methods of solving second order non homogenous differential equation with constant coefficients to solve higher order nonhomogeneous differential cquation with constant cocfficients. Example: Solve y-8y"+16y=-18sinx Solution A-87+16-0 = (-4)²-0=r-4=r=+2 Let y,-Acosx+Bsinx , y',--Asinx+Bcosx, y",= -Acosx-Bsinx y",- Asinx-Bcosx , y",- Acosx+Bsinx Acosx+Bsinx+8Acosx+8Bsinx+164cosx+16Bsinx-18sinx 25Acosx+25Bsinx=-18sinx 254-04-0 25B-18->В-18/25 y, =c,e* +cxe* +c;e* 18 sinx 25 a)Variation of parameters In this method, the particular solution y, has the form y,-v,u,+vuz+. +v,l, Where u1, uz, ..., U, are taken from ya=ciuj+c;uz+. +Cntty. To find v, vz, . V, we must solve the following linear equations. For v', v's ., v vu, + vu, +... +vu, =0 vu, + v,i", +... +v = 0 +... +vu a-3 +... +vu," f(x) (n-2) (n-2) vju, vu," =0 (n-I) (e-1)
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