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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 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" + 16y=0; y₁ = cos (4x), y₂ = sin(4x); y(0) = 9, y'(0) = 3 (...) Why is the function y₁ = cos(4x) a solution to the differential equation? Select the correct choice below and fill in the answer box to complete your choice. (Type an expression using x as the variable.) A. The function y₁ = cos (4x) is a solution because when the function and its second derivative, y₁" = - 16 cos (4x), are substituted into the equation, the result is a true statement. OB. The function y₁ = cos(4x) is a solution because when the function and its indefinite integral, are substituted into the equation, the result is a true statement. Why is the function y₂ = sin(4x) a solution to the differential equation? Select the correct choice below and fill in the answer box to complete your choice. A. The function y₂ = sin(4x) is a solution because when the function and its second derivative, y₂"= -16 sin(4x), are substituted into the equation, the result is a true statement. B. The function y₂ = sin(4x) 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) = 9 and y'(0) = 3 is y=

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 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" + 16y=0; y₁ = cos (4x), y₂ = sin(4x); y(0) = 9, y'(0) = 3 (...) Why is the function y₁ = cos(4x) a solution to the differential equation? Select the correct choice below and fill in the answer box to complete your choice. (Type an expression using x as the variable.) A. The function y₁ = cos (4x) is a solution because when the function and its second derivative, y₁" = - 16 cos (4x), are substituted into the equation, the result is a true statement. OB. The function y₁ = cos(4x) is a solution because when the function and its indefinite integral, are substituted into the equation, the result is a true statement. Why is the function y₂ = sin(4x) a solution to the differential equation? Select the correct choice below and fill in the answer box to complete your choice. A. The function y₂ = sin(4x) is a solution because when the function and its second derivative, y₂"= -16 sin(4x), are substituted into the equation, the result is a true statement. B. The function y₂ = sin(4x) 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) = 9 and y'(0) = 3 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 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'' + 16y=0; y₁ = cos (4x), y₂ = sin(4x); y(0) = 9, y'(0) = 3 Why is the function y₁ = cos(4x) a solution to the differential equation? Select the correct choice below and fill in the answer box to complete your choice. (Type an expression using x as the variable.) "" A. The function y₁ = cos(4x) is a solution because when the function and its second derivative, y₁ = - 16 cos (4x), are substituted into the equation, the result is a true statement. B. The function y₁ = cos (4x) is a solution because when the function and its indefinite integral, are substituted into the equation, the result is a true statement. Why is the function Y2₂ = sin(4x) a solution to the differential equation? Select the correct choice below and fill in the answer box to complete your choice. "1= A. The function y₂ = sin(4x) is a solution because when the function and its second derivative, y₂' B. The function y2 = sin(4x) is a solution because when the function and its indefinite integral, " 16 sin (4x), are substituted into the equation, the result is a true statement. 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) = 9 and y'(0) = 3 is y=
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 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'' + 16y=0; y₁ = cos (4x), y₂ = sin(4x); y(0) = 9, y'(0) = 3 Why is the function y₁ = cos(4x) a solution to the differential equation? Select the correct choice below and fill in the answer box to complete your choice. (Type an expression using x as the variable.) "" A. The function y₁ = cos(4x) is a solution because when the function and its second derivative, y₁ = - 16 cos (4x), are substituted into the equation, the result is a true statement. B. The function y₁ = cos (4x) is a solution because when the function and its indefinite integral, are substituted into the equation, the result is a true statement. Why is the function Y2₂ = sin(4x) a solution to the differential equation? Select the correct choice below and fill in the answer box to complete your choice. "1= A. The function y₂ = sin(4x) is a solution because when the function and its second derivative, y₂' B. The function y2 = sin(4x) is a solution because when the function and its indefinite integral, " 16 sin (4x), are substituted into the equation, the result is a true statement. 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) = 9 and y'(0) = 3 is y=
Show directly that the given functions are linearly dependent on the real line. That is, find a nontrivial linear combination of the following functions that vanishes identically. f(x) = 6x, g(x) = 4x², h(x) = 8x10x² Enter the non-trivial linear combination. (32)6x +4x²+) (8x-10x²) = 0
Transcribed Image Text:Show directly that the given functions are linearly dependent on the real line. That is, find a nontrivial linear combination of the following functions that vanishes identically. f(x) = 6x, g(x) = 4x², h(x) = 8x10x² Enter the non-trivial linear combination. (32)6x +4x²+) (8x-10x²) = 0
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