The coefficient matrix for a system of linear differential equations of the form

Burhan Hopper
2021-01-04
Answered

The coefficient matrix for a system of linear differential equations of the form

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Bella

Answered 2021-01-05
Author has **81** answers

By theorem 6.19 we know that the solution is

$y={c}_{1}{e}^{{\lambda}_{1}t}{u}_{1}+\dots +{c}_{n}{e}^{{\lambda}_{n}t}{u}_{n}$

with${\lambda}_{i}$ the eigenvalues of the matrix A nad u_i the eigenvectors
Thus for this case we then obtain the general solution:

$\left[\begin{array}{c}{y}_{1}\\ {y}_{2}\\ {y}_{3}\end{array}\right]=y={c}_{1}{e}^{3t}\left[\begin{array}{c}1\\ 1\\ 0\end{array}\right]+{c}_{2}{e}^{0t}\left[\begin{array}{c}1\\ 5\\ 1\end{array}\right]+{c}_{3}{e}^{0t}\left[\begin{array}{c}2\\ 1\\ 4\end{array}\right]$

Thus we obtain:

${y}_{1}={c}_{1}{e}^{3t}+{c}_{2}{e}^{0t}+2{c}_{3}{e}^{0t}={c}_{1}{e}^{3t}+{c}_{2}+2{c}_{3}$

${y}_{2}={c}_{1}{e}^{3t}+5{c}_{2}{e}^{0t}+{c}_{3}{e}^{0t}={c}_{1}{e}^{3t}+5{c}_{2}+{c}_{3}$

${y}_{3}={c}_{2}{e}^{0t}+4{c}_{3}{e}^{0t}={c}_{2}+4{c}_{3}$

with

Thus we obtain:

asked 2021-06-10

Determine whether the given set S is a subspace of the vector space V.

A. V=${P}_{5}$ , and S is the subset of ${P}_{5}$ consisting of those polynomials satisfying p(1)>p(0).

B.$V={R}_{3}$ , and S is the set of vectors $({x}_{1},{x}_{2},{x}_{3})$ in V satisfying ${x}_{1}-6{x}_{2}+{x}_{3}=5$ .

C.$V={R}^{n}$ , and S is the set of solutions to the homogeneous linear system Ax=0 where A is a fixed m×n matrix.

D. V=${C}^{2}(I)$ , and S is the subset of V consisting of those functions satisfying the differential equation y″−4y′+3y=0.

E. V is the vector space of all real-valued functions defined on the interval [a,b], and S is the subset of V consisting of those functions satisfying f(a)=5.

F. V=${P}_{n}$ , and S is the subset of ${P}_{n}$ consisting of those polynomials satisfying p(0)=0.

G.$V={M}_{n}(R)$ , and S is the subset of all symmetric matrices

A. V=

B.

C.

D. V=

E. V is the vector space of all real-valued functions defined on the interval [a,b], and S is the subset of V consisting of those functions satisfying f(a)=5.

F. V=

G.

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