Match the solution curve with one of the differential equations. x y"- 7y' +12y = 0 Ⓒy" + 4y = 0 Ⓒy" + y = 0 Oy" + 2y' + y = 0 y" - 5y' - 6y = 0 y" + 2y' + 2y = 0 X Explain your reasoning. (Assume that k, k₁, and k₂ are all positive.) The auxiliary equation should have two positive roots, so that the solution has the form c₁ek1* + c₂ek₂x. The differential equation should have the form y" + k²y = 0 where k = 1, so that the period of the solution is 21. The auxiliary equation should have one positive and one negative root, so that the solution has the form c₁e1* + c₂e-k₂x. The auxiliary equation should have a repeated negative root, so that the solution has the form c,e-x + exe k The differential equation should have the form y" + k²y = 0 where k = 2, so that the period of the solution is I. The auxiliary equation should have a pair of complex roots a ± ßi where a < 0, so that the solution has the form eax (c₁ cos(x) + C₂ sin(x)). X
Match the solution curve with one of the differential equations. x y"- 7y' +12y = 0 Ⓒy" + 4y = 0 Ⓒy" + y = 0 Oy" + 2y' + y = 0 y" - 5y' - 6y = 0 y" + 2y' + 2y = 0 X Explain your reasoning. (Assume that k, k₁, and k₂ are all positive.) The auxiliary equation should have two positive roots, so that the solution has the form c₁ek1* + c₂ek₂x. The differential equation should have the form y" + k²y = 0 where k = 1, so that the period of the solution is 21. The auxiliary equation should have one positive and one negative root, so that the solution has the form c₁e1* + c₂e-k₂x. The auxiliary equation should have a repeated negative root, so that the solution has the form c,e-x + exe k The differential equation should have the form y" + k²y = 0 where k = 2, so that the period of the solution is I. The auxiliary equation should have a pair of complex roots a ± ßi where a < 0, so that the solution has the form eax (c₁ cos(x) + C₂ sin(x)). X
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.
x
y" - 7y' + 12y =
0
Ⓒy" + 4y = 0
Ⓒy" + y = 0
Oy" + 2y' + y = 0
y" - 5y' - 6y = 0
y" + 2y + 2y = 0
X
Explain your reasoning. (Assume that k, k₁, and k₂ are all positive.)
The auxiliary equation should have two positive roots, so that the solution has the form
c₂₁ek₁x + ₂k₂x.
The differential equation should have the form y" + k²y = 0 where k = 1, so that the period of the solution is 21.
The auxiliary equation should have one positive and one negative root, so that the solution has the form c₁ek1* + c₂e-k₂x.
The auxiliary equation should have a repeated negative root, so that the solution has the form c₂e-x +
EXe k
The differential equation should have the form y" + k²y = 0 where k = 2, so that the period of the solution is I.
The auxiliary equation should have a pair of complex roots a ± ßi where a < 0, so that the solution has the form
ex (c₁ cos(x) + C₂ sin(x)).
X
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