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" - 4y =0; y₁ = e²x, y₂ = 2x; y(0) = 2, y'(0) = 0 Why is the function y₁ = ²x a solution to the differential equation? Select the correct choice below and fill in the answer box to complete your choice. A. The function y₁ = ²x is a solution because when the function and its second derivative, y₁"= 4e²x, are substituted into the equation, the result is a true statement. OB. The function y₁ = 2x 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 = -2x a solution to the differential equation? Select the correct choice below and fill in the answer box to complete your choice. The function y₂ =e-2x is a solution because when the function and its second derivative, y₂'' = 4e-2x, are substituted into the equation, the result is a true statement. OB. The function y₂ = 2x 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) = 2 and y'(0)=0 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" - 4y =0; y₁ = e²x, y₂ = 2x; y(0) = 2, y'(0) = 0 Why is the function y₁ = ²x a solution to the differential equation? Select the correct choice below and fill in the answer box to complete your choice. A. The function y₁ = ²x is a solution because when the function and its second derivative, y₁"= 4e²x, are substituted into the equation, the result is a true statement. OB. The function y₁ = 2x 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 = -2x a solution to the differential equation? Select the correct choice below and fill in the answer box to complete your choice. The function y₂ =e-2x is a solution because when the function and its second derivative, y₂'' = 4e-2x, are substituted into the equation, the result is a true statement. OB. The function y₂ = 2x 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) = 2 and y'(0)=0 is y=.
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: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'' - 4y =0; y₁ = e²x,
A.
Why is the function y₁ = ²x a solution to the differential equation? Select the correct choice below and fill in the answer box to complete your choice.
The function
B. The function y₁
A.
B.
Y₁
The function
₂ Y₂ =
Why is the function Y2 = e
Y2
= e
- 2x.
= e X; y(0) = 2, y'(0) = 0
2x
"1=
is a solution because when the function and its second derivative, y₁
= e
The function y₂ = e
= e2x is a solution because when the function and its indefinite integral, are substituted into the equation, the result is a true statement.
"
2x
4e²
- 2x
a solution to the differential equation? Select the correct choice below and fill in the answer box to complete your choice.
2x
is a solution because when the function and its second derivative, y2
"
are substituted into the equation, the result is a true statement.
"' = 4e
- 2x
are substituted into the equation, the result is a true statement.
2x
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) = 2 and y'(0) = 0 is y = .
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