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Given the third-order linear homogeneous differential equation: y (3)-y=0 Select all correct answers Hide answer choices Three linearly independent solutions of the given differential equation are: e², e-*/2.xe-x/2 Three linearly independent solutions of the given differential equation are: √√3 2 Ⓡee/2 sin (- (5). √3 2 x), e ¹/2 co cos ( -x) If y (0)=0, y '(0) = 1, y ''(0) = -1 then y=(-2) e (-1/2) sin (√3 x + x) 2

Given the third-order linear homogeneous differential equation: y (3)-y=0 Select all correct answers Hide answer choices Three linearly independent solutions of the given differential equation are: e², e-*/2.xe-x/2 Three linearly independent solutions of the given differential equation are: √√3 2 Ⓡee/2 sin (- (5). √3 2 x), e ¹/2 co cos ( -x) If y (0)=0, y '(0) = 1, y ''(0) = -1 then y=(-2) e (-1/2) sin (√3 x + x) 2

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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**C** Then \( y = \left(\frac{-2}{\sqrt{3}}\right) e^{-x/2} \sin\left(\frac{\sqrt{3}}{2} x + \pi\right) \) is a solution of the associated initial value problem. Three linearly independent solutions of the given differential equation are: **D** \( e^x, \, e^{-x/2} \sin\left(\frac{\sqrt{3}}{2} x\right), \, e^{-x/2} \cos\left(\frac{\sqrt{3}}{2} x\right) \) **E** \( y \equiv 0 \) is a solution of the given differential equation: \( y^{(3)} - y = 0 \) If \( y(0) = 0, \, y'(0) = 1, \, y''(0) = -1 \) **F** Then \( y = \left(\frac{2}{\sqrt{3}}\right) e^{-x/2} \sin\left(\frac{\sqrt{3}}{2} x\right) \) is a solution of the associated initial value problem.
Transcribed Image Text:**C** Then \( y = \left(\frac{-2}{\sqrt{3}}\right) e^{-x/2} \sin\left(\frac{\sqrt{3}}{2} x + \pi\right) \) is a solution of the associated initial value problem. Three linearly independent solutions of the given differential equation are: **D** \( e^x, \, e^{-x/2} \sin\left(\frac{\sqrt{3}}{2} x\right), \, e^{-x/2} \cos\left(\frac{\sqrt{3}}{2} x\right) \) **E** \( y \equiv 0 \) is a solution of the given differential equation: \( y^{(3)} - y = 0 \) If \( y(0) = 0, \, y'(0) = 1, \, y''(0) = -1 \) **F** Then \( y = \left(\frac{2}{\sqrt{3}}\right) e^{-x/2} \sin\left(\frac{\sqrt{3}}{2} x\right) \) is a solution of the associated initial value problem.
Given the third-order linear homogeneous differential equation: \[ y^{(3)} - y = 0 \] **Select all correct answers** Three linearly independent solutions of the given differential equation are: **A** \[ e^x, \, e^{-x/2}, \, x \, e^{-x/2} \] Three linearly independent solutions of the given differential equation are: **B** \[ e^x, \, e^{x/2} \sin\left(\frac{\sqrt{3}}{2}x\right), \, e^{x/2} \cos\left(\frac{\sqrt{3}}{2}x\right) \] \[ \text{If } y(0) = 0, \, y'(0) = 1, \, y''(0) = -1 \] **C** \[ \text{then } y = \left(\frac{-2}{\sqrt{3}}\right) e^{-x/2} \sin\left(\frac{\sqrt{3}}{2}x + \pi\right) \]
Transcribed Image Text:Given the third-order linear homogeneous differential equation: \[ y^{(3)} - y = 0 \] **Select all correct answers** Three linearly independent solutions of the given differential equation are: **A** \[ e^x, \, e^{-x/2}, \, x \, e^{-x/2} \] Three linearly independent solutions of the given differential equation are: **B** \[ e^x, \, e^{x/2} \sin\left(\frac{\sqrt{3}}{2}x\right), \, e^{x/2} \cos\left(\frac{\sqrt{3}}{2}x\right) \] \[ \text{If } y(0) = 0, \, y'(0) = 1, \, y''(0) = -1 \] **C** \[ \text{then } y = \left(\frac{-2}{\sqrt{3}}\right) e^{-x/2} \sin\left(\frac{\sqrt{3}}{2}x + \pi\right) \]
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