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A homogeneous second-order linear differential equation, two functions y, and y2, and a pair of initial conditions are given. First verify that y, and y₂ are solutions of the differential equation. Then find a particular solution of the form y=c₁y₁ + C₂y2 that satisfies the given initial conditions. Primes denote derivatives with respect to x. y" + 2y' +y=0; y₁ = e Y₂ = xey(0)=3, y'(0) = -5 Why is the function y₁ = e a solution to the differential equation? Select the correct choice below and fill in the answer box to complete your choice. OA. The function y₁ = ex is a solution because when the function and its indefinite integral, OB. The function y₁ = ex is a solution because when the function, its first derivative y₁' = are substituted into the equation, the result is a true statement. and its second derivative, y₁"= are substituted into the equation, the result is a true statement. Why is the function y₂ = xe a solution to the differential equation? Select the correct choice below and fill in the answer box to complete your choice. OA. The function y₂ = xe* is a solution because when the function, its derivative, y₂ =, and its second derivative, y₂" = are substituted into the equation, the result is a true statement. OB. The function y₂ = xe* is a solution because when the function and its indefinite integral, are substituted into the equation, the result is a true statement. The particular solution of the form y=c₁y₁ + C₂Y₂ that satisfies the initial conditions y(0) = 3 and y'(0) = -5 is y=

A homogeneous second-order linear differential equation, two functions y, and y2, and a pair of initial conditions are given. First verify that y, and y₂ are solutions of the differential equation. Then find a particular solution of the form y=c₁y₁ + C₂y2 that satisfies the given initial conditions. Primes denote derivatives with respect to x. y" + 2y' +y=0; y₁ = e Y₂ = xey(0)=3, y'(0) = -5 Why is the function y₁ = e a solution to the differential equation? Select the correct choice below and fill in the answer box to complete your choice. OA. The function y₁ = ex is a solution because when the function and its indefinite integral, OB. The function y₁ = ex is a solution because when the function, its first derivative y₁' = are substituted into the equation, the result is a true statement. and its second derivative, y₁"= are substituted into the equation, the result is a true statement. Why is the function y₂ = xe a solution to the differential equation? Select the correct choice below and fill in the answer box to complete your choice. OA. The function y₂ = xe* is a solution because when the function, its derivative, y₂ =, and its second derivative, y₂" = are substituted into the equation, the result is a true statement. OB. The function y₂ = xe* is a solution because when the function and its indefinite integral, are substituted into the equation, the result is a true statement. The particular solution of the form y=c₁y₁ + C₂Y₂ that satisfies the initial conditions y(0) = 3 and y'(0) = -5 is y=

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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A homogeneous second-order linear differential equation, two functions y₁ and y2, and a pair of initial conditions are given. First verify that y₁ and ₂ are solutions of the differential equation. Then find a particular solution of the form y=C₁y₁ + C₂y2 that satisfies the given initial conditions. Primes denote derivatives with respect to x. -X. y" + 2y'+y=0; y₁=ex, y₂ = xe; y(0) = 3, y'(0) = -5 ... -X Why is the function y₁ =e a solution to the differential equation? Select the correct choice below and fill in the answer box to complete your choice. OA. The function y₁ = e exis a solution because when the function and its indefinite integral, OB. The function y₁ = exis a solution because when the function, its first derivative y₁' = are substituted into the equation, the result is a true statement. and its second derivative, y₁ 11= -X OA. The function y₂ = xe is a solution because when the function, its derivative, y2´ = OB. The function y₂ = xex is a solution because when the function and its indefinite integral, " -X Why is the function y₂ = xe a solution to the differential equation? Select the correct choice below and fill in the answer box to complete your choice. are substituted into the equation, the result is a true statement. The particular solution of the form y = C₁Y₁ + C₂Y2 that satisfies the initial conditions y(0) = 3 and y'(0) = -5 is y = and its second derivative, y₂'' = are substituted into the equation, the result is a true statement. are substituted into the equation, the result is a true statement.
Transcribed Image Text:A homogeneous second-order linear differential equation, two functions y₁ and y2, and a pair of initial conditions are given. First verify that y₁ and ₂ are solutions of the differential equation. Then find a particular solution of the form y=C₁y₁ + C₂y2 that satisfies the given initial conditions. Primes denote derivatives with respect to x. -X. y" + 2y'+y=0; y₁=ex, y₂ = xe; y(0) = 3, y'(0) = -5 ... -X Why is the function y₁ =e a solution to the differential equation? Select the correct choice below and fill in the answer box to complete your choice. OA. The function y₁ = e exis a solution because when the function and its indefinite integral, OB. The function y₁ = exis a solution because when the function, its first derivative y₁' = are substituted into the equation, the result is a true statement. and its second derivative, y₁ 11= -X OA. The function y₂ = xe is a solution because when the function, its derivative, y2´ = OB. The function y₂ = xex is a solution because when the function and its indefinite integral, " -X Why is the function y₂ = xe a solution to the differential equation? Select the correct choice below and fill in the answer box to complete your choice. are substituted into the equation, the result is a true statement. The particular solution of the form y = C₁Y₁ + C₂Y2 that satisfies the initial conditions y(0) = 3 and y'(0) = -5 is y = and its second derivative, y₂'' = are substituted into the equation, the result is a true statement. are substituted into the equation, the result is a true statement.
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