Match the solution curve with one of the differential equations. O y" - 2y' – 3y = 0 y" + 2y' + y = 0 О y" - бу' + 8y %3D 0 y" + 4y = 0 O y" + y = 0 y" + 2y' + 2y = 0 Explain your reasoning. (Assume that k, k,, and k, are all positive.) O The auxiliary equation should have two positive roots, so that the solution has the form c,ek1* + c,ek2*. The auxiliary equation should have a pair of complex roots a ± ßi where a < 0, so that the solution has the form eax(c, cos(ßx) + C2 sin(ßx)). The differential equation should have the form y" + k
Match the solution curve with one of the differential equations. O y" - 2y' – 3y = 0 y" + 2y' + y = 0 О y" - бу' + 8y %3D 0 y" + 4y = 0 O y" + y = 0 y" + 2y' + 2y = 0 Explain your reasoning. (Assume that k, k,, and k, are all positive.) O The auxiliary equation should have two positive roots, so that the solution has the form c,ek1* + c,ek2*. The auxiliary equation should have a pair of complex roots a ± ßi where a < 0, so that the solution has the form eax(c, cos(ßx) + C2 sin(ßx)). The differential equation should have the form y" + k
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:Match the solution curve with one of the differential equations.
yA
O y" – 2y' – 3y = 0
О у" + 2y' + у %3D0
у" — бу' + 8у %3D 0
y" + 4y = 0
%3D
у" + у %3D 0
y" + 2y' + 2y = 0
%D
Explain your reasoning. (Assume that k, k,, and k, are all positive.)
The auxiliary equation should have two positive roots, so that the solution has the form c,eK1* + c,e*2*.
The auxiliary equation should have a pair of complex roots a + ßi where a < 0, so that the solution has the form
eax(c, cos(Bx) + c2 sin(ßx)).
The differential equation should have the form y" + k<y = 0 where k =
2, so that the period of the solution is T.
The auxiliary equation should have one positive and one negative root, so that the solution has the form
czeki* + cze-k2*.
The differential equation should have the form y" + ky = 0 where k =
1, so that the period of the solution is 2n.
%3D
-kx
The auxiliary equation should have a repeated negative root, so that the solution has the form c,eK* + C,xe¬KX.
-kx
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