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Consider the differential equation x²y" - 7xy' + 15y = 0; x³, x5, (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" - 7xy' + 15y = 0; x³, x5, (0, ∞) We are asked to verify that the solutions are linearly independent. That is, there do not exist constants c₁ and C₂, not both zero, such that c₁x³ + c₂x5 = 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. W(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) = x³ and f₂(x) = x5. Complete the Wronskian for functions. x3 x5 W(x³, x5)= f₁ f₂ 3x2

Consider the differential equation x²y" - 7xy' + 15y = 0; x³, x5, (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" - 7xy' + 15y = 0; x³, x5, (0, ∞) We are asked to verify that the solutions are linearly independent. That is, there do not exist constants c₁ and C₂, not both zero, such that c₁x³ + c₂x5 = 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. W(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) = x³ and f₂(x) = x5. Complete the Wronskian for functions. x3 x5 W(x³, x5)= f₁ f₂ 3x2

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter11: Differential Equations
Section11.1: Solutions Of Elementary And Separable Differential Equations
Problem 5E
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Consider the differential equation x²y" - 7xy' + 15y = 0; x³, x5, (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" - 7xy' + 15y = 0; x³, x5, (0, ∞) We are asked to verify that the solutions are linearly independent. That is, there do not exist constants c₁ and c₂, not both zero, such that c₁x³ + c₂x5 = 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 W(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) = x³ and f₂(x) = x5. Complete the Wronskian for these functions. x³ x5 w(x³, x5)= 3x²
Transcribed Image Text:Consider the differential equation x²y" - 7xy' + 15y = 0; x³, x5, (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" - 7xy' + 15y = 0; x³, x5, (0, ∞) We are asked to verify that the solutions are linearly independent. That is, there do not exist constants c₁ and c₂, not both zero, such that c₁x³ + c₂x5 = 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 W(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) = x³ and f₂(x) = x5. Complete the Wronskian for these functions. x³ x5 w(x³, x5)= 3x²
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