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Consider the differential equation x²y" - 7xy + 15y = 0; x³, x³, (0, ∞o). Verify that the given functions form a fundamental set of solutions of the differential equation on the indicated interval Form the general solution.

Consider the differential equation x²y" - 7xy + 15y = 0; x³, x³, (0, ∞o). Verify that the given functions form a fundamental set of solutions of the differential equation on the indicated interval Form the general solution.

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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Consider the differential equation x²y" - 7xy + 15y = 0; x³, x³, (0, ∞0). Verify that the given functions form a fundamental set of solutions of the differential equation on the indicated interval Form the general solution.
Transcribed Image Text:Consider the differential equation x²y" - 7xy + 15y = 0; x³, x³, (0, ∞0). Verify that the given functions form a fundamental set of solutions of the differential equation on the indicated interval Form the general solution.
We are given the following homogenous differential equation and pair of solutions on the given interval. x²y" - 7xy + 15y = 0; x³, x³, (0, m) We are asked to verify that the solutions are linearly independent. That is, there do not exist constants c, and c, not both zero, such that c,x+c 0. While this may be clear for these solutions that are different powers of x, we have a formal test to verify the linear independence. Recall the definition of the Wronskian for the case of two functions f, and f, each of which have a first derivative |f₁ f₂ If $₂ By Theorem 4.1.3, if W(f, f₂) = 0 for every x in the interval of the solution, then solutions are linearly independent Let f(x) = x² and f(x)=x. Complete the Wronskian for these functions. W(x³, x ³) = W(f₂, f₂) = 3x²
Transcribed Image Text:We are given the following homogenous differential equation and pair of solutions on the given interval. x²y" - 7xy + 15y = 0; x³, x³, (0, m) We are asked to verify that the solutions are linearly independent. That is, there do not exist constants c, and c, not both zero, such that c,x+c 0. While this may be clear for these solutions that are different powers of x, we have a formal test to verify the linear independence. Recall the definition of the Wronskian for the case of two functions f, and f, each of which have a first derivative |f₁ f₂ If $₂ By Theorem 4.1.3, if W(f, f₂) = 0 for every x in the interval of the solution, then solutions are linearly independent Let f(x) = x² and f(x)=x. Complete the Wronskian for these functions. W(x³, x ³) = W(f₂, f₂) = 3x²
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