Get answer: To get a feel for higher order ODEs. show that the given functions are

To get a feel for higher-order ODEs, show that the given functions are solutions and form a basis on any interval. Use Wronskians. (In Prob. 2, x > 0.) \(1, x, x^{2}, x^{3}, \quad y^{\text {iv }}=0\) Text Transcription: 1, x, x^2, x^3, y^iv = 0

To get a feel for higher-order ODEs, show that the given functions are solutions and form a basis on any interval. Use Wronskians. (In Prob. 2, x > 0.) \(1, x^{2}, x^{4}, \quad x^{2} y^{\prime \prime \prime}-3 x y^{\prime \prime}+3 y^{\prime}=0\) Text Transcription: 1, x^2, x^4, x^2 y”’ - 3xy” + 3y” = 0

To get a feel for higher-order ODEs, show that the given functions are solutions and form a basis on any interval. Use Wronskians. (In Prob. 2, x > 0.) \(e^{x}, x e^{x} \cdot x^{2} e^{x}\) \(y^{\prime \prime \prime}-3 y^{\prime \prime}+3 y^{\prime}-y=0\) Text Transcription: e^x, xe^x cdot x^2 e^x y”’ - 3y” + 3y’ - y = 0

To get a feel for higher-order ODEs, show that the given functions are solutions and form a basis on any interval. Use Wronskians. (In Prob. 2, x > 0.) \(e^{2 x} \cos x, e^{2 x} \sin x, e^{-2 x} \cos x, e^{-2 x} \sin x\), \(y^{\text {iv }}-6 y^{\prime \prime}+25 y=0\) Text Transcription: e^2x cos x, e^2x sin x, e^-2x cos x, e^-2x sin x, y^iv - 6y” + 25y = 0

To get a feel for higher-order ODEs, show that the given functions are solutions and form a basis on any interval. Use Wronskians. (In Prob. 2, x > 0.) \(1, x, \cos 3 x, \sin 3 x, \quad y^{\text {iv }}+9 y^{\prime \prime}=0\) Text Transcription: 1, x, cos 3x, sin 3x, y^iv + 9y” = 0

General Properties of Solutions of Linear ODEs.These properties are important in obtaining new solutions from given ones. Therefore extend Team Project 34 in Sec. 2.2 to nth-order ODEs. Explore statements on sums and multiples of solutions of (1) and (2) systematically and with proofs. Recognize clearly that no new ideas are needed in this extension from n = 2 to general n.

Are the given functions linearly independent or dependent on the positive x-axis? (Give a reason.) \(1, e^{x}, e^{-x}\) Text Transcription: 1, e^x, e^-x

Are the given functions linearly independent or dependent on the positive x-axis? (Give a reason.) x + 1, x + 2, x

Are the given functions linearly independent or dependent on the positive x-axis? (Give a reason.) \(\ln x, \ln x^{2},(\ln x)^{2}\) Text Transcription: ln x, ln x^2, ln x^2

Are the given functions linearly independent or dependent on the positive x-axis? (Give a reason.) \(e^{x}, e^{-x}, \sinh 2 x\) Text Transcription: e^x, e^-x, sinh 2x

Are the given functions linearly independent or dependent on the positive x-axis? (Give a reason.) \(x^{2}, x|x|, x\) Text Transcription: x^2, x | x |, x

Are the given functions linearly independent or dependent on the positive x-axis? (Give a reason.) x, 1/x, 0

Are the given functions linearly independent or dependent on the positive x-axis? (Give a reason.) sin 2x, sin x, cos x

Are the given functions linearly independent or dependent on the positive x-axis? (Give a reason.) \(\cos ^{2} x, \sin ^{2} x, \cos 2 x\) Text Transcription: cos^2 x, sin^2 x, cos 2x

Are the given functions linearly independent or dependent on the positive x-axis? (Give a reason.) tan x, cot x, 1

Are the given functions linearly independent or dependent on the positive x-axis? (Give a reason.) \((x-1)^{2},(x+1)^{2}, x\) Text Transcription: (x - 1)^2, (x + 1)^2, x

Are the given functions linearly independent or dependent on the positive x-axis? (Give a reason.) \(\sin x, \sin \frac{1}{2} x\) Text Transcription: sin x, sin 1 / 2 x

Are the given functions linearly independent or dependent on the positive x-axis? (Give a reason.) \(\cosh x, \sinh x, \cosh ^{2} x\) Text Transcription: cosh x, sinh x, cosh^2 x

Are the given functions linearly independent or dependent on the positive x-axis? (Give a reason.) \(\cos ^{2} x, \sin ^{2} x, 2 \pi\) Text Transcription: cos^2 x, sin^2 x, 2 pi

Linear Independence and Dependence. (a) Investigate the given question about a set S of functions on an interval I . Give an example. Prove your answer. (1) If S contains the zero function, can S be linearly independent? (2) If S is linearly independent on a subinterval J of I. is it linearly independent on I? (3) If S is linearly dependent on a subinterval J of I, is it linearly dependent on I? (4) If S is linearly independent on I, is it linearly independent on a subinterval J? (5) If S is linearly dependent on I. is it linearly independent on a subinterval J? (6) If S is linearly dependent on I, and if T contains S, is T linearly dependent on I? (b) In what cases can you use the Wronskian for testing linear independence? By what other means can you perform such a test?