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Show that y, (x) = e 3* and y,(x) = e 4x are linearly independent on I = (-0, ∞) and find a second order homogeneous equation having the pair as a fundemental set of solutions. y" -y'+12y =0 b) О у" +3у' — 4у-0 c) O y" - 8y ' +4y=0 d) Оу" +4y'- 12 у -0 y" -y'- 12y=0 f) O None of the above.

Show that y, (x) = e 3* and y,(x) = e 4x are linearly independent on I = (-0, ∞) and find a second order homogeneous equation having the pair as a fundemental set of solutions. y" -y'+12y =0 b) О у" +3у' — 4у-0 c) O y" - 8y ' +4y=0 d) Оу" +4y'- 12 у -0 y" -y'- 12y=0 f) O None of the above.

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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Show that y, (x) = e 3* and y,(x) = e -4x are linearly independent on I= (-00, ∞) and find a second order homogeneous equation having the pair as a fundemental set of solutions. y" -y'+12y =0 b) y" + 3y'- 4y =0 c) O y " – 8y '+4y=0 d) O y" +4y '– 12y =0 - y " -y'- 12y=0 f) O None of the above.
Transcribed Image Text:Show that y, (x) = e 3* and y,(x) = e -4x are linearly independent on I= (-00, ∞) and find a second order homogeneous equation having the pair as a fundemental set of solutions. y" -y'+12y =0 b) y" + 3y'- 4y =0 c) O y " – 8y '+4y=0 d) O y" +4y '– 12y =0 - y " -y'- 12y=0 f) O None of the above.
Expert Solution
Step 1

We know if an Auxiliary equation of a second order differential equation is 

(m-3)(m+4)=0            (such that m=3, -4)

then we have two linearly interdepend solutions y1=e3x and y2=e-4x

Since Auxiliary equation is

       (m-3)(m+4)=0⇔ m2+4m-3m-12=0⇔ m2+m-12=0

Hence required second order differential equation is

⇔ (D2+D-12)y=0⇔ D2y+Dy-12y=0⇔y"+y'-12y=0

Hence none of them is correct

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