Consider the differential equation x²y" - 4xy' + 6y = 0; x², 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. Step 1 We are given the following homogenous differential equation and pair of solutions on the given interval. x²y" - 4xy' + 6y = 0; x², x³, (0, ∞0) We are asked to verify that the solutions are linearly independent. That is, there do not exist constants c₁ and ₂, not both zero, such that c₁x² + ₂x³ = 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 f₁ and f₂, each of which have a first derivative. f₁ f₂ f₁ f₂ # By Theorem 4.1.3, if W(f₁, f₂) = 0 for every x in the interval of the solution, then solutions are linearly independent. Let f₁(x) = x² and f₂(x) = x³. Complete the Wronskian for these functions. |x² W(f₁, f₂) = w(x², x³) = 2x
Consider the differential equation x²y" - 4xy' + 6y = 0; x², 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. Step 1 We are given the following homogenous differential equation and pair of solutions on the given interval. x²y" - 4xy' + 6y = 0; x², x³, (0, ∞0) We are asked to verify that the solutions are linearly independent. That is, there do not exist constants c₁ and ₂, not both zero, such that c₁x² + ₂x³ = 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 f₁ and f₂, each of which have a first derivative. f₁ f₂ f₁ f₂ # By Theorem 4.1.3, if W(f₁, f₂) = 0 for every x in the interval of the solution, then solutions are linearly independent. Let f₁(x) = x² and f₂(x) = x³. Complete the Wronskian for these functions. |x² W(f₁, f₂) = w(x², x³) = 2x
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:Consider the differential equation
x²y" - 4xy' + 6y = 0;
Verify that the given functions form a fundamental set of solutions of the differential equation on the indicated
interval.
Form the general solution.
Step 1
We are given the following homogenous differential equation and pair of solutions on the given interval.
x²y" - 4xy' + 6y = 0; x², x³, (0, 0)
We are asked to verify that the solutions are linearly independent. That is, there do not exist constants C₁ and C₂¹
not both zero, such that c₁x² + ₂x³ = 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 f₁ and f2, each of which have a first derivative.
f₁ f₂
f₁' f₂
By Theorem 4.1.3, if W(f₁, f₂) = 0 for every x in the interval of the solution, then solutions are linearly independent.
#
w(f₁, f₂) =
x², x³, (0, ∞0).
Let f₁(x) = x² and f₂(x) = x³. Complete the Wronskian for these functions.
|x²
x³
w(x², x³) =
2x
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