MAT244H1 Study Guide - Final Guide: Saddle Point, Integral Curve, Phase Plane

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Published on 22 Apr 2016
School
UTSG
Department
Mathematics
Course
MAT244H1
Professor
MAT244: Introduction to ODEs
summary of the last topic
1 Eigenvector Reduction Method
The general solution to the system
y
S
=Ay
S
(1)
Case 1. If λ1
λ2then use formula
y
S
(x) = eλ1xa1v
S
1+eλ2xa2v
S
2,(2)
where v
S
1, v
S
2are eigenvectors. Use the following alternative formula if you do not want
calculate eigenvectors
y
S
(x) = eλ1xy0
S
+eλ2xeλ1x
λ2λ2
(Aλ1I)y0
S
.(3)
Case 2. If λ1=λ2=λthen use the formula
y
S
(x) = a1eλxv
S
1+eλx(a2x v
S
1+a2v
S
2).(4)
Use the following alternative formula if you do not want calculate eigenvectors
y
S
(x) = eλx[y
S
0+x(AλI)y
S
0].(5)
Case 3. If λ=σ+then use the formula
y
S
(x) = eσxcos (ωx)y
S
0+sin (ωx)
ω(AσI)y
S
0.(6)
—————————————————————————————————————
Example 1. Find the general solution of the following systems
i.
y
S
=2 1
41y
S
ii.
y
S
=62
82y
S
iii.
y
S
=1 4
11y
S
Solution. For the first system we have
p(λ) = λ2λ6=0λ1=2, λ2= 3,
Eigenvector Reduction Method 1
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with eigenvectors v
S
1=1
4, and v
S
2=1
1. Therefore, the general solution is
y
S
(x) = e2xc11
4+e3xc21
1.
For the second system, we have
p(λ) = λ24λ+ 4 = 0 λ= 2.
Let us use the formula (5) and obtain
y
S
(x) = e2x
c1
c2+x42
84c1
c2=e2x
c1+ 2x(2c1c2)
c2+ 4x(2c1c2)
For the last system, we have
p(λ) = λ2+ 2λ+ 5 = 0 λ=1 + 2i.
We use formula (6) and write
y
S
(x) = ex
cos (2x)c1
c2+sin (2x)
20 4
1 0 c1
c2=ex
cos (2x)c1
c2+sin (2x)
24c2
c1
————————————————————————————————————
2 Fundamental matrix
The fundamental matrix of the system y
S
=Ay
S
Case 1. If λ1
λ2and v
S
1, v
S
2are eigenvectors then for Q= [v1
S
|v2
S
]we have
Φ(x) = Q
eλ1x0
0eλ2x!Q1.(7)
Case 2. If λ1=λ2=λand v1
S
is the only one eigenvector, then find v2through the
equation (AλI)v
S
2=v
S
1and for Q= [v1
S
|v2
S
]we have
Φ(x) = eλxQ1x
0 1 Q1.(8)
Case 3. If λ=σ+then for Q= [i(v1
S
v
S
2)|v1
S
+v
S
2]we have
Φ(x) = eσxQcos (ωx)sin (ωx)
sin (ωx)cos (ωx)Q1.(9)
———————————————————————————————————————–
Example 2. Let us find the fundamental matrix of the system in the above example. For
the first one, we have Q=1 1
4 1 and then
Φ(x) =1 1
4 1 e2x0
0e3x!1 1
4 1 1
.
2Section 2
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Document Summary

Mat244: introduction to odes summary of the last topic. If 1 (cid:2) 2 then use formula ys (x) = e 1xa1vs 1 + e 2xa2vs 2, 1 (1) (2) where vs 1, vs 2 are eigenvectors. Use the following alternative formula if you do not want calculate eigenvectors ys (x) = e 1xy0s + e 2x e 1x. If 1 = 2 = then use the formula ys (x) = a1e x vs 1 + e x(a2 x vs 1 + a2 vs 2). Use the following alternative formula if you do not want calculate eigenvectors ys (x) = e x[ys 0 + x(a i)ys 0]. If = + i then use the formula ys (x) = e x(cid:20) cos ( x)ys 0 + sin ( x) (a i)ys 0(cid:21). Find the general solution of the following systems (3) (4) (5) (6) i. ii. iii. ys ys ys.

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