5t? (t) = and x (t) 10t Consider the vectors x' (2) (a) Compute the Wronskian of x and x2) . W = (b) In what intervals are x and x) linearly independent? CO. D = (c) What conclusion can be drawn about coefficients in the system of homogeneous differential equations satisfied by x and x2)? Choose one of the coefficients of the ODE in stadard form must be discontinuous at to and to %D %3D (d) Find the system of equations x' = P(t)x. P(t) =
5t? (t) = and x (t) 10t Consider the vectors x' (2) (a) Compute the Wronskian of x and x2) . W = (b) In what intervals are x and x) linearly independent? CO. D = (c) What conclusion can be drawn about coefficients in the system of homogeneous differential equations satisfied by x and x2)? Choose one of the coefficients of the ODE in stadard form must be discontinuous at to and to %D %3D (d) Find the system of equations x' = P(t)x. P(t) =
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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Question
![Consider the vectors \( \mathbf{x}^{(1)}(t) = \begin{pmatrix} 5t^2 \\ 10t \end{pmatrix} \) and \( \mathbf{x}^{(2)}(t) = \begin{pmatrix} e^t \\ e^t \end{pmatrix} \).
**(a)** Compute the Wronskian of \( \mathbf{x}^{(1)} \) and \( \mathbf{x}^{(2)} \).
\[ W = \boxed{\hspace{2cm}} \]
**(b)** In what intervals are \( \mathbf{x}^{(1)} \) and \( \mathbf{x}^{(2)} \) linearly independent?
\[ D = (\boxed{\hspace{0.5cm}}, \boxed{\hspace{0.5cm}}) \cup (\boxed{\hspace{0.5cm}}, \boxed{\hspace{0.5cm}}), \]
**(c)** What conclusion can be drawn about coefficients in the system of homogeneous differential equations satisfied by \( \mathbf{x}^{(1)} \) and \( \mathbf{x}^{(2)} \)?
\[ \text{Choose one} \, \boxed{\text{Select}} \, \text{of the coefficients of the ODE in standard form} \]
must be discontinuous at \( t_0 = \boxed{\hspace{0.5cm}} \) and \( t_0 = \boxed{\hspace{0.5cm}} \).
**(d)** Find the system of equations \( \mathbf{x}' = \mathbf{P}(t)\mathbf{x} \).
\[ \mathbf{P}(t) = \begin{pmatrix}
\boxed{\hspace{0.5cm}} & \boxed{\hspace{0.5cm}} \\
\boxed{\hspace{0.5cm}} & \boxed{\hspace{0.5cm}}
\end{pmatrix} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcde309f6-a963-407f-aa62-f36631bd17cf%2Fabbe3552-3c86-423d-99a3-8f5b58add633%2Fukkerlu_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Consider the vectors \( \mathbf{x}^{(1)}(t) = \begin{pmatrix} 5t^2 \\ 10t \end{pmatrix} \) and \( \mathbf{x}^{(2)}(t) = \begin{pmatrix} e^t \\ e^t \end{pmatrix} \).
**(a)** Compute the Wronskian of \( \mathbf{x}^{(1)} \) and \( \mathbf{x}^{(2)} \).
\[ W = \boxed{\hspace{2cm}} \]
**(b)** In what intervals are \( \mathbf{x}^{(1)} \) and \( \mathbf{x}^{(2)} \) linearly independent?
\[ D = (\boxed{\hspace{0.5cm}}, \boxed{\hspace{0.5cm}}) \cup (\boxed{\hspace{0.5cm}}, \boxed{\hspace{0.5cm}}), \]
**(c)** What conclusion can be drawn about coefficients in the system of homogeneous differential equations satisfied by \( \mathbf{x}^{(1)} \) and \( \mathbf{x}^{(2)} \)?
\[ \text{Choose one} \, \boxed{\text{Select}} \, \text{of the coefficients of the ODE in standard form} \]
must be discontinuous at \( t_0 = \boxed{\hspace{0.5cm}} \) and \( t_0 = \boxed{\hspace{0.5cm}} \).
**(d)** Find the system of equations \( \mathbf{x}' = \mathbf{P}(t)\mathbf{x} \).
\[ \mathbf{P}(t) = \begin{pmatrix}
\boxed{\hspace{0.5cm}} & \boxed{\hspace{0.5cm}} \\
\boxed{\hspace{0.5cm}} & \boxed{\hspace{0.5cm}}
\end{pmatrix} \]
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