Write the second order inhomogeneous linear differential equation (t) + a₁ (t)*(t) + ao(t)x(t) = f(t) (5) as a two-dimensional system x = Ax+ g. Let {₁, ₂} be a fundamental set of solutions for the homogeneous equation (t) + a₁(t)x(t) + ao(t)x(t) = 0. By using the method of variation of parameters for systems, show that Xp(t)=¹u₂(t)u₁(T) – u₁(t)u₂(T) f(T) dt, W[u1, U₂] (T) is a solution of (5). Hence show that *(t) + x(t) = f(t)
Write the second order inhomogeneous linear differential equation (t) + a₁ (t)*(t) + ao(t)x(t) = f(t) (5) as a two-dimensional system x = Ax+ g. Let {₁, ₂} be a fundamental set of solutions for the homogeneous equation (t) + a₁(t)x(t) + ao(t)x(t) = 0. By using the method of variation of parameters for systems, show that Xp(t)=¹u₂(t)u₁(T) – u₁(t)u₂(T) f(T) dt, W[u1, U₂] (T) is a solution of (5). Hence show that *(t) + x(t) = f(t)
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.6: Applications And The Perron-frobenius Theorem
Problem 69EQ: Let x=x(t) be a twice-differentiable function and consider the second order differential equation...
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Hi, this is not a graded question and the (5) just mean the equation(this also shown in the question)
![3. Write the second order inhomogeneous linear differential equation
x(t) + a₁ (t)x(t) + ao(t)x(t) = f(t)
(5)
as a two-dimensional system x = Ax+g. Let {₁, ₂} be a fundamental set of solutions
for the homogeneous equation (t) + a₁(t)x(t) + ao(t)x(t) = 0. By using the method of
variation of parameters for systems, show that
Xp(t) = f* ¹2(t)ur (T) – U₁(t)u2(7),
W[u₁, U₂] (T)
is a solution of (5). Hence show that
has
as a particular solution.
*(t) + x(t) = f(t)
Xp (t) = sin(
f(T) dt,
sin(t-T)f(T) dt](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F490cbcd2-ad81-426b-824f-903aced284ba%2Fe3b90f5f-74df-463f-8741-850054ea0dac%2Ff8wgqbr_processed.jpeg&w=3840&q=75)
Transcribed Image Text:3. Write the second order inhomogeneous linear differential equation
x(t) + a₁ (t)x(t) + ao(t)x(t) = f(t)
(5)
as a two-dimensional system x = Ax+g. Let {₁, ₂} be a fundamental set of solutions
for the homogeneous equation (t) + a₁(t)x(t) + ao(t)x(t) = 0. By using the method of
variation of parameters for systems, show that
Xp(t) = f* ¹2(t)ur (T) – U₁(t)u2(7),
W[u₁, U₂] (T)
is a solution of (5). Hence show that
has
as a particular solution.
*(t) + x(t) = f(t)
Xp (t) = sin(
f(T) dt,
sin(t-T)f(T) dt
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