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The indicated function y₁(x) is a solution of the given differential equation. Use reduction of order or formula (5) in Section 4.2, e-SP(x) dx (5) x²(x) Y₂ Y₂ = Y₁(x) [² as instructed, to find a second solution y₂(x). 6y"+y¹ - y = 0; Y₁ = ex/3 11 dx

The indicated function y₁(x) is a solution of the given differential equation. Use reduction of order or formula (5) in Section 4.2, e-SP(x) dx (5) x²(x) Y₂ Y₂ = Y₁(x) [² as instructed, to find a second solution y₂(x). 6y"+y¹ - y = 0; Y₁ = ex/3 11 dx

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 16E: Find the general solution for each differential equation. Verify that each solution satisfies the...
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### Reduction of Order: Finding a Second Solution The indicated function \(y_1(x)\) is a solution of the given differential equation. Use reduction of order or formula (5) in Section 4.2, \[ y_2 = y_1(x) \int \frac{e^{-\int P(x) \, dx}}{y_1^2(x)} \, dx \tag{5} \] as instructed, to find a second solution \(y_2(x)\). Given differential equation and initial solution: \[ 6y'' + y' - y = 0; \quad y_1 = e^{x/3} \] ### Solution \[ y_2 = \boxed{\phantom{asd}} \] By applying the given method, we can find the second solution to complement the initial solution \(y_1(x)\).
Transcribed Image Text:### Reduction of Order: Finding a Second Solution The indicated function \(y_1(x)\) is a solution of the given differential equation. Use reduction of order or formula (5) in Section 4.2, \[ y_2 = y_1(x) \int \frac{e^{-\int P(x) \, dx}}{y_1^2(x)} \, dx \tag{5} \] as instructed, to find a second solution \(y_2(x)\). Given differential equation and initial solution: \[ 6y'' + y' - y = 0; \quad y_1 = e^{x/3} \] ### Solution \[ y_2 = \boxed{\phantom{asd}} \] By applying the given method, we can find the second solution to complement the initial solution \(y_1(x)\).
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