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3. [20 points: (a) 5, (b) 15] For a 3 x 3 system of ODEs, $\vec{x}\,' = A\vec{x}$ where A is an 3 × 3 matrix, the general solution is given by $\vec{x}(t) = c_1\vec{x}_1(t) + c_2\vec{x}_2(t) + c_3\vec{x}_3(t)$ where $\vec{x}_1, \vec{x}_2, \vec{x}_3$ form a fundamental set of solutions (are solutions to ODE and linearly independent). (a) Suppose A has 3 simple eigenvalues, $\lambda_1, \lambda_2, \lambda_3$ with associated eigenvectors $\vec{v}_1, \vec{v}_2, \vec{v}_3$ respectively. Show that $\vec{x}_1 = e^{\lambda_1 t}\vec{v}_1$, $\vec{x}_2 = e^{\lambda_2 t}\vec{v}_2$, $\vec{x}_3 = e^{\lambda_3 t}\vec{v}_3$, form a fundamental set of solutions. (b) Using part (a), solve the following system of IVPs: $\vec{x}\,' = \begin{pmatrix} -2 & 1 & 0\\0 & 2 & 1\\0 & 1 & 2 \end{pmatrix} \vec{x}$, $\vec{x}(0) = \begin{pmatrix} -1\\7\\13 \end{pmatrix}$ Hint: Eigenvalues of the matrix will be distinct, real, and integers.

          3. [20 points: (a) 5, (b) 15]
For a 3 x 3 system of ODEs, $\vec{x}\,' = A\vec{x}$ where A is an 3 × 3 matrix, the general solution is given by
$\vec{x}(t) = c_1\vec{x}_1(t) + c_2\vec{x}_2(t) + c_3\vec{x}_3(t)$
where $\vec{x}_1, \vec{x}_2, \vec{x}_3$ form a fundamental set of solutions (are solutions to ODE and linearly independent).
(a) Suppose A has 3 simple eigenvalues, $\lambda_1, \lambda_2, \lambda_3$ with associated eigenvectors $\vec{v}_1, \vec{v}_2, \vec{v}_3$ respectively.
Show that
$\vec{x}_1 = e^{\lambda_1 t}\vec{v}_1$,
$\vec{x}_2 = e^{\lambda_2 t}\vec{v}_2$,
$\vec{x}_3 = e^{\lambda_3 t}\vec{v}_3$,
form a fundamental set of solutions.
(b) Using part (a), solve the following system of IVPs:
$\vec{x}\,' = \begin{pmatrix} -2 & 1 & 0\\0 & 2 & 1\\0 & 1 & 2 \end{pmatrix} \vec{x}$, $\vec{x}(0) = \begin{pmatrix} -1\\7\\13 \end{pmatrix}$
Hint: Eigenvalues of the matrix will be distinct, real, and integers.

3. [20 points: (a) 5, (b) 15]
For a 3 x 3 system of ODEs, x⃗ ' = Ax⃗ where A is an 3 × 3 matrix, the general solution is given by
x⃗(t) = c1x⃗1(t) + c2x⃗2(t) + c3x⃗3(t)
where x⃗1, x⃗2, x⃗3 form a fundamental set of solutions (are solutions to ODE and linearly independent).
(a) Suppose A has 3 simple eigenvalues, λ1, λ2, λ3 with associated eigenvectors v⃗1, v⃗2, v⃗3 respectively.
Show that
x⃗1 = e^λ1 tv⃗1,
x⃗2 = e^λ2 tv⃗2,
x⃗3 = e^λ3 tv⃗3,
form a fundamental set of solutions.
(b) Using part (a), solve the following system of IVPs:
x⃗ ' =
< p m a t r i x >
x⃗, x⃗(0) =
< p m a t r i x >
Hint: Eigenvalues of the matrix will be distinct, real, and integers.

Added by Kristy M.

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