Question 1

Consider the matrix   Compute B-1

Accepted Answer

Question 2

Consider the matrix  
Which of these is a complete list of eigenvalue-eigenvector pairs of A?

Accepted Answer

A)  $ \lambda_{1}=4, \quad \xi_{1}=\left(\begin{array}{l}3 \ 1\end{array}\right) \quad \lambda_{2}=-4, \quad \xi_{2}=\left(\begin{array}{c}-1 \ 1\end{array}\right) $ 
B)  $ \lambda_{1}=4, \xi_{1}=\left(\begin{array}{l}1 \ 3\end{array}\right), \lambda_{2}=-4, \xi_{2}=\left(\begin{array}{c}1 \ -1\end{array}\right) $ 
C)  $ \lambda_{1}=2, \xi_{1}=\left(\begin{array}{l}3 \ 1\end{array}\right) \quad \lambda_{2}=-2, \quad \xi_{2}=\left(\begin{array}{c}-1 \ 1\end{array}\right) $ 
D)  $ \lambda_{1}=2, \quad \xi_{1}=\left(\begin{array}{l}1 \ 3\end{array}\right) \quad \lambda_{2}=-2, \quad \xi_{2}=\left(\begin{array}{c}1 \ -1\end{array}\right) $ 
A)  $ \lambda_{1}=4, \quad \xi_{1}=\left(\begin{array}{l}3 \ 1\end{array}\right) \quad \lambda_{2}=-4, \quad \xi_{2}=\left(\begin{array}{c}-1 \ 1\end{array}\right) $ 
B)  $ \lambda_{1}=4, \xi_{1}=\left(\begin{array}{l}1 \ 3\end{array}\right), \lambda_{2}=-4, \xi_{2}=\left(\begin{array}{c}1 \ -1\end{array}\right) $ 
C)  $ \lambda_{1}=2, \xi_{1}=\left(\begin{array}{l}3 \ 1\end{array}\right) \quad \lambda_{2}=-2, \quad \xi_{2}=\left(\begin{array}{c}-1 \ 1\end{array}\right) $ 
D)  $ \lambda_{1}=2, \quad \xi_{1}=\left(\begin{array}{l}1 \ 3\end{array}\right) \quad \lambda_{2}=-2, \quad \xi_{2}=\left(\begin{array}{c}1 \ -1\end{array}\right) $ A

Question 3

Consider this system of first-order differential equations:
   
(i) Transform this system into a second-order differential equation whose solution is X1 .
   
(ii) Find the general solution of the differential equation in part (i).
(iii) Use your solution in (ii) to now find X2.

Accepted Answer

(i) D

Question 4

Consider the first-order homogeneous system of linear differential equations
   
The fundamental matrix of this system,   (t), is invertible.

Accepted Answer

A) True 
 B)False

Question 5

Consider the first-order homogeneous system of linear differential equations
  
What is the general solution of this system? Here,   and   are arbitrary real constants.

Accepted Answer

A)  $ \mathbf{x}(t)=C_{1} e^{-2 t}\left[\begin{array}{l}\sin (-8 t) \ \cos (-8 t)\end{array}\right)+C_{2} e^{-2 t}\left[\begin{array}{l}-\cos (-8 t) \ \sin (-8 t)\end{array}\right) $ 
B)  $ \mathbf{x}(t)=C_{1} e^{8 t}\left(\begin{array}{l}\sin (2 t) \ \cos (2 t)\end{array}\right)+C_{2} e^{8 t}\left(\begin{array}{l}-\cos (2 t) \ \sin (2 t)\end{array}\right) $ 
C) $ \mathbf{x}(t)=C_{1} e^{-8 t}\left[\begin{array}{l}\sin (2 t) \ \cos (2 t)\end{array}\right]+C_{2} e^{-8 t}\left[\begin{array}{l}-\cos (2 t) \ \sin (2 t)\end{array}\right) $ 
D)  $ \mathbf{x}(t)=C_{1} e^{2 t}\left(\begin{array}{l}\sin (-8 t) \ \cos (-8 t)\end{array}\right)+C_{2} e^{2 t}\left[\begin{array}{l}-\cos (-8 t) \ \sin (-8 t)\end{array}\right) $ 
A)  $ \mathbf{x}(t)=C_{1} e^{-2 t}\left[\begin{array}{l}\sin (-8 t) \ \cos (-8 t)\end{array}\right)+C_{2} e^{-2 t}\left[\begin{array}{l}-\cos (-8 t) \ \sin (-8 t)\end{array}\right) $ 
B)  $ \mathbf{x}(t)=C_{1} e^{8 t}\left(\begin{array}{l}\sin (2 t) \ \cos (2 t)\end{array}\right)+C_{2} e^{8 t}\left(\begin{array}{l}-\cos (2 t) \ \sin (2 t)\end{array}\right) $ 
C) $ \mathbf{x}(t)=C_{1} e^{-8 t}\left[\begin{array}{l}\sin (2 t) \ \cos (2 t)\end{array}\right]+C_{2} e^{-8 t}\left[\begin{array}{l}-\cos (2 t) \ \sin (2 t)\end{array}\right) $ 
D)  $ \mathbf{x}(t)=C_{1} e^{2 t}\left(\begin{array}{l}\sin (-8 t) \ \cos (-8 t)\end{array}\right)+C_{2} e^{2 t}\left[\begin{array}{l}-\cos (-8 t) \ \sin (-8 t)\end{array}\right) $

Question 6

Transform this nonhomogeneous second-order initial value problem into an initial value problem comprised of two first-order differential equations:

Accepted Answer

A)   
B)  . 
C)  .. 
D)  ... 
A)   
B)  . 
C)  .. 
D)  ...

Question 7

Consider the first-order homogeneous system of linear differential equations
  
Given a fundamental matrix   (t) for the system, which of these is the general solution of this system? Here,   is an arbitrary constant vector.

Accepted Answer

A)  x(t) =   (t)C 
B)  x(t) =   (t) + C 
C)  x(t) =   -1(t)C 
D)  x(t) =   (0)C 
A)  x(t) =   (t)C 
B)  x(t) =   (t) + C 
C)  x(t) =   -1(t)C 
D)  x(t) =   (0)C

Question 8

Consider the first-order homogeneous system of linear differential equations
 
Which of these is the general solution of the system? Here, C1 and C2 are arbitrary real constants.

Accepted Answer

A)  $ \mathbf{x}(t)=C_{1}\left(\begin{array}{l}3 \ -10\end{array}\right) e^{-2 t}+C_{2}\left(\begin{array}{l}0 \ 1\end{array}\right) e^{-5 t} $ 
B)  $ \mathbf{x}(t)=C_{1}\left(\begin{array}{l}3 \ -10\end{array}\right) e^{-2 t}+C_{2}\left(\begin{array}{c}0 \ -1\end{array}\right) e^{5 t} $ 
C)  $ \mathbf{x}(t)=C_{1}\left(\begin{array}{l}3 \ -10\end{array}\right) e^{-2 t}+C_{2}\left(\begin{array}{l}-10 \ -3\end{array}\right) e^{2 t} $ 
D)  $ \mathbf{x}(t)=C_{1}\left(\begin{array}{l}-1 \ 0\end{array}\right) e^{5 t}+C_{2}\left(\begin{array}{l}3 \ -10\end{array}\right) e^{-2 t} $ 
E)  $ \mathbf{x}(t)=C_{1}\left(\begin{array}{l}0 \ 1\end{array}\right) e^{-5 t}+C_{2}\left(\begin{array}{l}-10 \ -3\end{array}\right) e^{2 t} $ 
A)  $ \mathbf{x}(t)=C_{1}\left(\begin{array}{l}3 \ -10\end{array}\right) e^{-2 t}+C_{2}\left(\begin{array}{l}0 \ 1\end{array}\right) e^{-5 t} $ 
B)  $ \mathbf{x}(t)=C_{1}\left(\begin{array}{l}3 \ -10\end{array}\right) e^{-2 t}+C_{2}\left(\begin{array}{c}0 \ -1\end{array}\right) e^{5 t} $ 
C)  $ \mathbf{x}(t)=C_{1}\left(\begin{array}{l}3 \ -10\end{array}\right) e^{-2 t}+C_{2}\left(\begin{array}{l}-10 \ -3\end{array}\right) e^{2 t} $ 
D)  $ \mathbf{x}(t)=C_{1}\left(\begin{array}{l}-1 \ 0\end{array}\right) e^{5 t}+C_{2}\left(\begin{array}{l}3 \ -10\end{array}\right) e^{-2 t} $ 
E)  $ \mathbf{x}(t)=C_{1}\left(\begin{array}{l}0 \ 1\end{array}\right) e^{-5 t}+C_{2}\left(\begin{array}{l}-10 \ -3\end{array}\right) e^{2 t} $

Question 9

Consider the first-order homogeneous system of linear differential equations
  =   x
Select all of the correct eigenvalue-eigenvector pairs from the following choices.

Accepted Answer

A)  $ \lambda=9, \xi=\left(\begin{array}{l}1 \ 1\end{array}\right) $ 
B)  $ \lambda=-9, \xi=\left(\begin{array}{c}-1 \ 1\end{array}\right) $ 
C)  $ \lambda=4, \xi=\left(\begin{array}{l}-9 \ 4\end{array}\right) $ 
D)  $ \lambda=0, \xi=\left(\begin{array}{l}1 \ 0\end{array}\right) $ 
E)  $ \lambda=-4, \xi=\left(\begin{array}{l}9 \ -4\end{array}\right) $ 
A)  $ \lambda=9, \xi=\left(\begin{array}{l}1 \ 1\end{array}\right) $ 
B)  $ \lambda=-9, \xi=\left(\begin{array}{c}-1 \ 1\end{array}\right) $ 
C)  $ \lambda=4, \xi=\left(\begin{array}{l}-9 \ 4\end{array}\right) $ 
D)  $ \lambda=0, \xi=\left(\begin{array}{l}1 \ 0\end{array}\right) $ 
E)  $ \lambda=-4, \xi=\left(\begin{array}{l}9 \ -4\end{array}\right) $

Question 10

Consider the first-order homogeneous system of linear differential equations
  
Which of these is the fundamental matrix   (t) for this system?

Accepted Answer

A) $ \psi(t)=\left(\begin{array}{ll}-e^{-4 t} \sin (7 t) & e^{-4 t} \cos (7 t) \ e^{-4 t} \cos (7 t) & -e^{-4 t} \sin (7 t)\end{array}\right) $ 
B)  $ \psi(t)=\left\{\begin{array}{ll}e^{4 t} \sin (7 t) & -e^{4 t} \cos (7 t) \ -e^{4 t} \cos (7 t) & e^{4 t} \sin (7 t)\end{array}\right) $ 
C)  $ \psi(t)=\left(\begin{array}{ll}e^{4 t} \sin (7 t) & -e^{4 t} \cos (7 t) \ e^{4 t} \cos (7 t) & e^{4 t} \sin (7 t)\end{array}\right) $ 
D)  $ \psi(t)=\left(\begin{array}{ll}e^{-4 t} \sin (7 t) & -e^{-4 t} \cos (7 t) \ e^{-4 t} \cos (7 t) & e^{-4 t} \sin (7 t)\end{array}\right) $ 
A) $ \psi(t)=\left(\begin{array}{ll}-e^{-4 t} \sin (7 t) & e^{-4 t} \cos (7 t) \ e^{-4 t} \cos (7 t) & -e^{-4 t} \sin (7 t)\end{array}\right) $ 
B)  $ \psi(t)=\left\{\begin{array}{ll}e^{4 t} \sin (7 t) & -e^{4 t} \cos (7 t) \ -e^{4 t} \cos (7 t) & e^{4 t} \sin (7 t)\end{array}\right) $ 
C)  $ \psi(t)=\left(\begin{array}{ll}e^{4 t} \sin (7 t) & -e^{4 t} \cos (7 t) \ e^{4 t} \cos (7 t) & e^{4 t} \sin (7 t)\end{array}\right) $ 
D)  $ \psi(t)=\left(\begin{array}{ll}e^{-4 t} \sin (7 t) & -e^{-4 t} \cos (7 t) \ e^{-4 t} \cos (7 t) & e^{-4 t} \sin (7 t)\end{array}\right) $

Question 11

Consider the first-order homogeneous system of linear differential equations
  
Which of these is the fundamental matrix   (t) for this system?

Accepted Answer

A)  $ \psi(t)=\left(\begin{array}{l}4 e^{12 t}-4 e^{-12 t} \ 3 e^{12 t}-3 e^{-12 t}\end{array}\right) $ 
B)  $ \Psi(t)=\left(\begin{array}{l}4 e^{12 t}-4 e^{-12 t} \ 3 e^{12 t} 3 e^{-12 t}\end{array}\right) $ 
C)  $ \psi(t)=\left(\begin{array}{ll}9 e^{-16 t} & -9 e^{9 t} \ 16 e^{-16 t} & 16 e^{9 t}\end{array}\right) $ 
D)  $ \psi(t)=\left(\begin{array}{ll}9 e^{-16 t} & 9 e^{9 t} \ -16 e^{-16 t} & 16 e^{9 t}\end{array}\right) $ 
A)  $ \psi(t)=\left(\begin{array}{l}4 e^{12 t}-4 e^{-12 t} \ 3 e^{12 t}-3 e^{-12 t}\end{array}\right) $ 
B)  $ \Psi(t)=\left(\begin{array}{l}4 e^{12 t}-4 e^{-12 t} \ 3 e^{12 t} 3 e^{-12 t}\end{array}\right) $ 
C)  $ \psi(t)=\left(\begin{array}{ll}9 e^{-16 t} & -9 e^{9 t} \ 16 e^{-16 t} & 16 e^{9 t}\end{array}\right) $ 
D)  $ \psi(t)=\left(\begin{array}{ll}9 e^{-16 t} & 9 e^{9 t} \ -16 e^{-16 t} & 16 e^{9 t}\end{array}\right) $

Question 12

Consider the first-order homogeneous system of linear differential equations
   
 All solution trajectories spiral towards the origin as t $\rightarrow$$\infty$.

Accepted Answer

A) True 
 B)False

Question 13

Consider the first-order homogeneous system of linear differential equations 
   
What is the general solution of this system?

Accepted Answer

The answer of Consider the first-order homogeneous system of linear...

Question 14

If A is an 2 × 4 matrix and B is an 4 × 9 matrix, then:

Accepted Answer

A)  BA is defined and has order 2 × 9. 
B)  BA is defined and has order 9 × 2. 
C)  AB is defined and has order 9 × 2. 
D)  AB is defined and has order 2 × 9. 
E)  Neither AB nor BA is defined. 
A)  BA is defined and has order 2 × 9. 
B)  BA is defined and has order 9 × 2. 
C)  AB is defined and has order 9 × 2. 
D)  AB is defined and has order 2 × 9. 
E)  Neither AB nor BA is defined.

Question 15

Consider the first-order nonhomogeneous initial value problem
 
Given a fundamental matrix   (t) for the system, what is the solution of this system if it is equipped with the initial condition x(0) = X0?

Accepted Answer

Question 16

Consider the first-order homogeneous system of linear differential equations
‪   
(i) For what real values of α does this system have complex eigenvalues?
(ii) What do the solution trajectories look like for the values of α found in part (i)?

Accepted Answer

(i) Any real number

Question 17

Consider the first-order homogeneous system of linear differential equations
 
And the following four vector functions:
 
Which of the following statements are true? Select all that apply.

Accepted Answer

A)    is a solution of this system, for all real numbers C1 ,C2 ,C3 , and C4 . 
B)  W [   (t),   (t)] ? 0 for every real number t. 
C)  5.5   + 4.5   + C is a solution of this system, for any real number C. 
D)  4   is a solution of this system. 
E)  {   ,   } is a fundamental set of solutions for this system. 
A)    is a solution of this system, for all real numbers C1 ,C2 ,C3 , and C4 . 
B)  W [   (t),   (t)] ? 0 for every real number t. 
C)  5.5   + 4.5   + C is a solution of this system, for any real number C. 
D)  4   is a solution of this system. 
E)  {   ,   } is a fundamental set of solutions for this system.

Question 18

Compute:

Accepted Answer

A)  $ \left[\begin{array}{ll}3 & -4 \ 7 & -9 \ -1 & -2\end{array}\right] $ 
B)  $ \left[\begin{array}{cc}9 & 0 \ 19 & -19 \ -3 & -2\end{array}\right] $ 
C)  $ \left[\begin{array}{ll}-3 & -8 \ -5 & 1 \ 1 & -2\end{array}\right] $ 
D)  $ \left[\begin{array}{ll}-9 & -12 \ -17 & 11 \ 3 & -2\end{array}\right] $ 
A)  $ \left[\begin{array}{ll}3 & -4 \ 7 & -9 \ -1 & -2\end{array}\right] $ 
B)  $ \left[\begin{array}{cc}9 & 0 \ 19 & -19 \ -3 & -2\end{array}\right] $ 
C)  $ \left[\begin{array}{ll}-3 & -8 \ -5 & 1 \ 1 & -2\end{array}\right] $ 
D)  $ \left[\begin{array}{ll}-9 & -12 \ -17 & 11 \ 3 & -2\end{array}\right] $

Question 19

Consider the first-order nonhomogeneous system of linear differential equations
 
Where the components of g(t) are continuous functions.Given a fundamental matrix   (t) for the system, what is the solution of this system if it is equipped with the initial condition x(3.6) = X 0?

Accepted Answer

A)  $ x(t)=\psi(0) \mathbf{x}_{0}+\psi(t) \int_{0}^{t} \psi^{-1}(s) \mathbf{g}(s) d s $ 
B)  $ x(t)=\psi(t) \psi^{-1}(3.6) \mathbf{x}_{0}+\psi(t) \int_{3.6}^{t} \psi^{-1}(s) g(s) d s $ 
C)  $ x(t)=\psi(3.6) \mathbf{x}_{0}+\psi(t) \int_{3.6}^{t} \psi^{-1}(s) \mathbf{g}(s) d s $ 
D)  $ x(t)=\psi(t) \psi^{-1}(0) \mathbf{x}_{0}+\psi(t) \int_{0}^{t} \psi^{-1}(s) g(s) d s $ 
A)  $ x(t)=\psi(0) \mathbf{x}_{0}+\psi(t) \int_{0}^{t} \psi^{-1}(s) \mathbf{g}(s) d s $ 
B)  $ x(t)=\psi(t) \psi^{-1}(3.6) \mathbf{x}_{0}+\psi(t) \int_{3.6}^{t} \psi^{-1}(s) g(s) d s $ 
C)  $ x(t)=\psi(3.6) \mathbf{x}_{0}+\psi(t) \int_{3.6}^{t} \psi^{-1}(s) \mathbf{g}(s) d s $ 
D)  $ x(t)=\psi(t) \psi^{-1}(0) \mathbf{x}_{0}+\psi(t) \int_{0}^{t} \psi^{-1}(s) g(s) d s $

Question 20

Consider the first-order homogeneous system of linear differential equations
  =   x
Which of these are eigenvalues for this system? Select all that apply.

Accepted Answer

A)  -4 + 7i 
B)  7 
C)  -7i 
D)  -4 
E)  7i
F) 7 + 4i
G) 7 - 4i
H) -4 - 7i 
A)  -4 + 7i 
B)  7 
C)  -7i 
D)  -4 
E)  7i
F) 7 + 4i
G) 7 - 4i
H) -4 - 7i

Consider the matrix Compute B^-1

Consider the matrix Which of these is a complete list of eigenvalue-eigenvector pairs of A?

Consider this system of first-order differential equations: (i) Transform this system into a second-order differential equation whose solution is X₁ . (ii) Find the general solution of the differential equation in part (i). (iii) Use your solution in (ii) to now find X₂.

Consider the first-order homogeneous system of linear differential equations The fundamental matrix of this system, (t), is invertible.

Consider the first-order homogeneous system of linear differential equations What is the general solution of this system? Here, and are arbitrary real constants.

Transform this nonhomogeneous second-order initial value problem into an initial value problem comprised of two first-order differential equations:

Consider the first-order homogeneous system of linear differential equations Given a fundamental matrix (t) for the system, which of these is the general solution of this system? Here, is an arbitrary constant vector.

Consider the first-order homogeneous system of linear differential equations Which of these is the general solution of the system? Here, C₁ and C₂ are arbitrary real constants.

Consider the first-order homogeneous system of linear differential equations = x Select all of the correct eigenvalue-eigenvector pairs from the following choices.

Consider the first-order homogeneous system of linear differential equations Which of these is the fundamental matrix (t) for this system?

Consider the first-order homogeneous system of linear differential equations Which of these is the fundamental matrix (t) for this system?

Consider the first-order homogeneous system of linear differential equations What is the general solution of this system?

If A is an 2 × 4 matrix and B is an 4 × 9 matrix, then:

Consider the first-order nonhomogeneous initial value problem Given a fundamental matrix (t) for the system, what is the solution of this system if it is equipped with the initial condition x(0) = X₀?

Consider the first-order homogeneous system of linear differential equations ‪ (i) For what real values of α does this system have complex eigenvalues? (ii) What do the solution trajectories look like for the values of α found in part (i)?

Consider the first-order homogeneous system of linear differential equations And the following four vector functions: Which of the following statements are true? Select all that apply.

Compute:

Consider the first-order nonhomogeneous system of linear differential equations Where the components of g(t) are continuous functions.Given a fundamental matrix (t) for the system, what is the solution of this system if it is equipped with the initial condition x(3.6) = X ₀?

Consider the first-order homogeneous system of linear differential equations = x Which of these are eigenvalues for this system? Select all that apply.

Introduction

First-Order Differential Equations

Second-Order Linear Differential Equations

Higher-Order Linear Differential Equations

Series Solutions of Second-Order Linear Equations

The Laplace Transform

Numerical Methods

Nonlinear Differential Equations and Stability

Partial Differential Equations and Fourier Series

Boundary Value Problems and Sturm-Liouville Theory

Filters

Exam 7: Systems of First-Order Linear Equations

Consider the matrix Compute B-1

Consider the matrix Which of these is a complete list of eigenvalue-eigenvector pairs of A?

Consider this system of first-order differential equations: (i) Transform this system into a second-order differential equation whose solution is X1 . (ii) Find the general solution of the differential equation in part (i). (iii) Use your solution in (ii) to now find X2.

Consider the first-order homogeneous system of linear differential equations The fundamental matrix of this system, (t), is invertible.

Consider the first-order homogeneous system of linear differential equations What is the general solution of this system? Here, and are arbitrary real constants.

Transform this nonhomogeneous second-order initial value problem into an initial value problem comprised of two first-order differential equations:

Consider the first-order homogeneous system of linear differential equations Given a fundamental matrix (t) for the system, which of these is the general solution of this system? Here, is an arbitrary constant vector.

Consider the first-order homogeneous system of linear differential equations Which of these is the general solution of the system? Here, C1 and C2 are arbitrary real constants.

Consider the first-order homogeneous system of linear differential equations = x Select all of the correct eigenvalue-eigenvector pairs from the following choices.

Consider the first-order homogeneous system of linear differential equations Which of these is the fundamental matrix (t) for this system?

Consider the first-order homogeneous system of linear differential equations Which of these is the fundamental matrix (t) for this system?

Consider the first-order homogeneous system of linear differential equations All solution trajectories spiral towards the origin as t →\rightarrow→∞\infty∞ .

Consider the first-order homogeneous system of linear differential equations What is the general solution of this system?

If A is an 2 × 4 matrix and B is an 4 × 9 matrix, then:

Consider the first-order nonhomogeneous initial value problem Given a fundamental matrix (t) for the system, what is the solution of this system if it is equipped with the initial condition x(0) = X0?

Consider the first-order homogeneous system of linear differential equations ‪ (i) For what real values of α does this system have complex eigenvalues? (ii) What do the solution trajectories look like for the values of α found in part (i)?

Consider the first-order homogeneous system of linear differential equations And the following four vector functions: Which of the following statements are true? Select all that apply.

Compute:

Consider the first-order nonhomogeneous system of linear differential equations Where the components of g(t) are continuous functions.Given a fundamental matrix (t) for the system, what is the solution of this system if it is equipped with the initial condition x(3.6) = X 0?

Consider the first-order homogeneous system of linear differential equations = x Which of these are eigenvalues for this system? Select all that apply.

Introduction

First-Order Differential Equations

Second-Order Linear Differential Equations

Higher-Order Linear Differential Equations

Series Solutions of Second-Order Linear Equations

The Laplace Transform

Numerical Methods

Nonlinear Differential Equations and Stability

Partial Differential Equations and Fourier Series

Boundary Value Problems and Sturm-Liouville Theory

Filters

Consider the matrix Compute B^-1

Consider this system of first-order differential equations: (i) Transform this system into a second-order differential equation whose solution is X₁ . (ii) Find the general solution of the differential equation in part (i). (iii) Use your solution in (ii) to now find X₂.

Consider the first-order homogeneous system of linear differential equations Which of these is the general solution of the system? Here, C₁ and C₂ are arbitrary real constants.

Consider the first-order nonhomogeneous initial value problem Given a fundamental matrix (t) for the system, what is the solution of this system if it is equipped with the initial condition x(0) = X₀?

Consider the first-order nonhomogeneous system of linear differential equations Where the components of g(t) are continuous functions.Given a fundamental matrix (t) for the system, what is the solution of this system if it is equipped with the initial condition x(3.6) = X ₀?