Mixing
Prediction methods to resolve differential equation.
$begingroup$
How resolved this difference equation used prediction methods?
$$X' = left[begin{array}{ccc}1&-1&-2\1&3&2\1&-1&2end{array}right]X + left[begin{array}{c}t^{2}\t+2\2end{array}right]$$
Calculate eigenvalues and eigenvectors:
$$lambda_{1} = 2 + 2i$$
$$lambda_{2} = 2-2i$$
$$lambda_{3} = 2$$
$$v_{1} = (i,-i,1)$$
$$v_{2} = (-i,i,1)$$
$$v_{3} = (-1,-1,1)$$
So,
$$X_{b} = C_{1}left(begin{array}{c}i\-i\iend{array}right)e^{(2+2i)t} + C_{2} left(begin{array}{c}-i\i\1 end{array}right)e^{(2-2i)t} + C_{3} left(begin{array}{c}1\-1\ 1end{array}right)e^{2t}$$
How continue?
ordinary-differential-equations systems-of-equations matrix-equations
edited Jan 8 at 2:00
bjcolby15
1,21611016
asked Jan 7 at 23:27
svssvs
6
$endgroup$
add a comment |
$begingroup$
How resolved this difference equation used prediction methods?
$$X' = left[begin{array}{ccc}1&-1&-2\1&3&2\1&-1&2end{array}right]X + left[begin{array}{c}t^{2}\t+2\2end{array}right]$$
Calculate eigenvalues and eigenvectors:
$$lambda_{1} = 2 + 2i$$
$$lambda_{2} = 2-2i$$
$$lambda_{3} = 2$$
$$v_{1} = (i,-i,1)$$
$$v_{2} = (-i,i,1)$$
$$v_{3} = (-1,-1,1)$$
So,
$$X_{b} = C_{1}left(begin{array}{c}i\-i\iend{array}right)e^{(2+2i)t} + C_{2} left(begin{array}{c}-i\i\1 end{array}right)e^{(2-2i)t} + C_{3} left(begin{array}{c}1\-1\ 1end{array}right)e^{2t}$$
How continue?
ordinary-differential-equations systems-of-equations matrix-equations
edited Jan 8 at 2:00
bjcolby15
1,21611016
asked Jan 7 at 23:27
svssvs
6
$endgroup$
add a comment |
$begingroup$
How resolved this difference equation used prediction methods?
$$X' = left[begin{array}{ccc}1&-1&-2\1&3&2\1&-1&2end{array}right]X + left[begin{array}{c}t^{2}\t+2\2end{array}right]$$
Calculate eigenvalues and eigenvectors:
$$lambda_{1} = 2 + 2i$$
$$lambda_{2} = 2-2i$$
$$lambda_{3} = 2$$
$$v_{1} = (i,-i,1)$$
$$v_{2} = (-i,i,1)$$
$$v_{3} = (-1,-1,1)$$
So,
$$X_{b} = C_{1}left(begin{array}{c}i\-i\iend{array}right)e^{(2+2i)t} + C_{2} left(begin{array}{c}-i\i\1 end{array}right)e^{(2-2i)t} + C_{3} left(begin{array}{c}1\-1\ 1end{array}right)e^{2t}$$
How continue?
ordinary-differential-equations systems-of-equations matrix-equations
edited Jan 8 at 2:00
bjcolby15
1,21611016
asked Jan 7 at 23:27
svssvs
6
$endgroup$
How resolved this difference equation used prediction methods?
$$X' = left[begin{array}{ccc}1&-1&-2\1&3&2\1&-1&2end{array}right]X + left[begin{array}{c}t^{2}\t+2\2end{array}right]$$
Calculate eigenvalues and eigenvectors:
$$lambda_{1} = 2 + 2i$$
$$lambda_{2} = 2-2i$$
$$lambda_{3} = 2$$
$$v_{1} = (i,-i,1)$$
$$v_{2} = (-i,i,1)$$
$$v_{3} = (-1,-1,1)$$
So,
$$X_{b} = C_{1}left(begin{array}{c}i\-i\iend{array}right)e^{(2+2i)t} + C_{2} left(begin{array}{c}-i\i\1 end{array}right)e^{(2-2i)t} + C_{3} left(begin{array}{c}1\-1\ 1end{array}right)e^{2t}$$
How continue?
ordinary-differential-equations systems-of-equations matrix-equations
ordinary-differential-equations systems-of-equations matrix-equations
edited Jan 8 at 2:00
bjcolby15
1,21611016
asked Jan 7 at 23:27
svssvs
6
edited Jan 8 at 2:00
bjcolby15
1,21611016
asked Jan 7 at 23:27
svssvs
6
edited Jan 8 at 2:00
bjcolby15
1,21611016
edited Jan 8 at 2:00
bjcolby15
1,21611016
edited Jan 8 at 2:00
bjcolby15
1,21611016
1,21611016
asked Jan 7 at 23:27
svssvs
6
asked Jan 7 at 23:27
svssvs
6
asked Jan 7 at 23:27
svssvs
6
6
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Hint: $$begin {matrix} dfrac {e^{i x} + e^{-i x}}{2} = ? text { and } dfrac {e^{i x} - e^{-i x}}{2} = ? end {matrix}$$
(in terms of what functions?)
Also, $$begin {matrix} dfrac {e^{x} + e^{-x}}{2} = ? text { and } dfrac {e^{x} - e^{-x}}{2} = ? end {matrix}$$
(in terms of what other functions?)
answered Jan 8 at 0:57
bjcolby15bjcolby15
1,21611016
$endgroup$
$begingroup$
$$ frac{e^{x} + e^{-x}}{2} = cosh{x} $$ $$ frac{e^{x} - e^{-x}}{2} = sinh{x} $$ $$ frac{e^{ix} + e^{-ix}}{2} = cos{x} $$ $$ frac{e^{ix} - e^{-ix}}{2i} = sin{x} $$ but, what next?
$endgroup$
– svs
Jan 8 at 12:54
add a comment |
Your Answer
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1 Answer
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1 Answer
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$begingroup$
Hint: $$begin {matrix} dfrac {e^{i x} + e^{-i x}}{2} = ? text { and } dfrac {e^{i x} - e^{-i x}}{2} = ? end {matrix}$$
(in terms of what functions?)
Also, $$begin {matrix} dfrac {e^{x} + e^{-x}}{2} = ? text { and } dfrac {e^{x} - e^{-x}}{2} = ? end {matrix}$$
(in terms of what other functions?)
answered Jan 8 at 0:57
bjcolby15bjcolby15
1,21611016
$endgroup$
$begingroup$
$$ frac{e^{x} + e^{-x}}{2} = cosh{x} $$ $$ frac{e^{x} - e^{-x}}{2} = sinh{x} $$ $$ frac{e^{ix} + e^{-ix}}{2} = cos{x} $$ $$ frac{e^{ix} - e^{-ix}}{2i} = sin{x} $$ but, what next?
$endgroup$
– svs
Jan 8 at 12:54
add a comment |
$begingroup$
Hint: $$begin {matrix} dfrac {e^{i x} + e^{-i x}}{2} = ? text { and } dfrac {e^{i x} - e^{-i x}}{2} = ? end {matrix}$$
(in terms of what functions?)
Also, $$begin {matrix} dfrac {e^{x} + e^{-x}}{2} = ? text { and } dfrac {e^{x} - e^{-x}}{2} = ? end {matrix}$$
(in terms of what other functions?)
answered Jan 8 at 0:57
bjcolby15bjcolby15
1,21611016
$endgroup$
$begingroup$
$$ frac{e^{x} + e^{-x}}{2} = cosh{x} $$ $$ frac{e^{x} - e^{-x}}{2} = sinh{x} $$ $$ frac{e^{ix} + e^{-ix}}{2} = cos{x} $$ $$ frac{e^{ix} - e^{-ix}}{2i} = sin{x} $$ but, what next?
$endgroup$
– svs
Jan 8 at 12:54
add a comment |
$begingroup$
Hint: $$begin {matrix} dfrac {e^{i x} + e^{-i x}}{2} = ? text { and } dfrac {e^{i x} - e^{-i x}}{2} = ? end {matrix}$$
(in terms of what functions?)
Also, $$begin {matrix} dfrac {e^{x} + e^{-x}}{2} = ? text { and } dfrac {e^{x} - e^{-x}}{2} = ? end {matrix}$$
(in terms of what other functions?)
answered Jan 8 at 0:57
bjcolby15bjcolby15
1,21611016
$endgroup$
Hint: $$begin {matrix} dfrac {e^{i x} + e^{-i x}}{2} = ? text { and } dfrac {e^{i x} - e^{-i x}}{2} = ? end {matrix}$$
(in terms of what functions?)
Also, $$begin {matrix} dfrac {e^{x} + e^{-x}}{2} = ? text { and } dfrac {e^{x} - e^{-x}}{2} = ? end {matrix}$$
(in terms of what other functions?)
answered Jan 8 at 0:57
bjcolby15bjcolby15
1,21611016
edited Jan 8 at 2:41
answered Jan 8 at 0:57
bjcolby15bjcolby15
1,21611016
answered Jan 8 at 0:57
bjcolby15bjcolby15
1,21611016
answered Jan 8 at 0:57
bjcolby15bjcolby15
1,21611016
1,21611016
$begingroup$
$$ frac{e^{x} + e^{-x}}{2} = cosh{x} $$ $$ frac{e^{x} - e^{-x}}{2} = sinh{x} $$ $$ frac{e^{ix} + e^{-ix}}{2} = cos{x} $$ $$ frac{e^{ix} - e^{-ix}}{2i} = sin{x} $$ but, what next?
$endgroup$
– svs
Jan 8 at 12:54
add a comment |
$begingroup$
$$ frac{e^{x} + e^{-x}}{2} = cosh{x} $$ $$ frac{e^{x} - e^{-x}}{2} = sinh{x} $$ $$ frac{e^{ix} + e^{-ix}}{2} = cos{x} $$ $$ frac{e^{ix} - e^{-ix}}{2i} = sin{x} $$ but, what next?
$endgroup$
– svs
Jan 8 at 12:54
$begingroup$
$$ frac{e^{x} + e^{-x}}{2} = cosh{x} $$ $$ frac{e^{x} - e^{-x}}{2} = sinh{x} $$ $$ frac{e^{ix} + e^{-ix}}{2} = cos{x} $$ $$ frac{e^{ix} - e^{-ix}}{2i} = sin{x} $$ but, what next?
$endgroup$
– svs
Jan 8 at 12:54
$begingroup$
$$ frac{e^{x} + e^{-x}}{2} = cosh{x} $$ $$ frac{e^{x} - e^{-x}}{2} = sinh{x} $$ $$ frac{e^{ix} + e^{-ix}}{2} = cos{x} $$ $$ frac{e^{ix} - e^{-ix}}{2i} = sin{x} $$ but, what next?
$endgroup$
– svs
Jan 8 at 12:54
add a comment |
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