8/8/2019 Euler Cauchy Equation
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Euler-Cauchy Equation
Muhammad Nadeem
8/8/2019 Euler Cauchy Equation
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Differential
Equations
ODE PDE
Second orderFirst order
LinearExact
Separable
Muhammad Nadeem
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NonhomogeneousHomogeneous
8/8/2019 Euler Cauchy Equation
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Second Order Homogeneous linear
equation: Euler-Cauchy Equation0
2=++ byyaxyx (1)
m
xy =Let its solution isSo (1) will be
=m
0)1(122
=++ mmm
bxamxxxmmx
Muhammad Nadeem
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Or
It is called characteristic or auxiliary equation. It will have two roots say
(2)
2
4)1()1(2
4)1()1(2
2
2
1baamandbaam =+=
These roots can be real and distinct, real double, or complex conjugate
0)1(2
=++ bmam
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Case I
Real and Distinct Roots
If pm =1 and qm =2So there will be two linearly independent solutions of equation 1
p=
q=
And General solutionqp
xcxcy 21 +=
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8/8/2019 Euler Cauchy Equation
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Example
Solution
Find the G.S. of the ODE 022
=+ yyxyx
mxy =
0}2)1({ =+m
xmmm
Let its solution is
So (1) will be 02)1(122
=+
mmm
xxmxxmmx
Muhammad Nadeem
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21 =m 22 =mSo general solution
2
2
2
1
+= xcxcy
022 =m
8/8/2019 Euler Cauchy Equation
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Example
Solution
Find the G.S. of the ODE
mxy =
0}312)1(4{ =++m
xmmm
Let its solution is
So (1) will be 0312)1(4122
=++
mmm
xxmxxmmx
031242
=++ yyxyx
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2/11 =m 2/32 =m
So general solution2/3
2
2/1
1
+= xcxcy
0384 2 =++ mm
8/8/2019 Euler Cauchy Equation
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Case II
Real Double RootIf 2/)1(21 amm ==
So there is only one solution2/)1(
1
axy
=
The second linearly independent solution (needed for basis)can be found by Reduction of Order. Let
Not G.S.!
=
2y
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112 yuyuy += 1112 2 yuyuyuy ++=
Since is solution of Eq. (1) so
0)()2(111111
2=+++++ buyyuyuaxyuyuyux
2
y
8/8/2019 Euler Cauchy Equation
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0)()2( 1112
111
2=+++++ byyaxyxuayyxxuyux
2/)1(
1
xaxy
=Since is solution of eq. (1) so
0111
2=++ byyaxyx 1112 yayyx =+and
So 0112
=+ xyuyux
0Since 2 =
2/)1(
2)(ln
a
xxy
=
And General solution of equation (1) will be
( ) 2/)1(212211 lna
xxccycycy
+=+=
Or xu ln=Hence second lineally independent solution of eq. (1) is
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Example
Solution
Find the G.S. of the ODE
mxy =
0}97)1({ =++m
xmmm
Let its solution is
So (1) will be 097)1(
122
=++
mmm
xxmxxmmx
0972
=++ yyxyx
Muhammad NadeemSEECS-NUST
321 == mm
So general solution3
21 )ln(
+= xxccy
0962 =++ mm
8/8/2019 Euler Cauchy Equation
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Example
Solution
Find the G.S. of the ODE
mxy =
0}2524)1(4{ =++m
xmmm
Let its solution is
So (1) will be 02524)1(4
122
=++
mmm
xxmxxmmx
0252442
=++ yyxyx
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2/521 == mm
So general solution2/5
21 )ln(
+= xxccy
025204 2 =++ mm
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Case III
Complex Conjugate RootsIf iqpmiqpm =+= 21 ,
Now we can find real solutions as
ln1 pxiqpiqpm +
21 ,mm
xxSo two complex solutions
And complex general solution iqpiqp xcxcy + += 21
xqxxx
y pmm
lncos2
21
1
=+
=
)lnsinln(cosln2 xqixqxexxx pxiqpiqp
m===
xqxi
xxy p
mm
lnsin2
21
2
=
=
So real general solution is
( )xqcxqcxy p lnsinlncos 21 +=Muhammad Nadeem
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Example
Solution
Find the G.S. of the ODE
mxy =
0}1)1({ =++m
xmmm
Let its solution is
So (1) will be 0)1(
122
=++
mmm
xxmxxmmx
02
=++ yyxyx
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im =1
So general solution
012
=+m
im =2
( )xqcxqcxy p lnsinlncos 21 +=
( )xcxcy lnsinlncos 21 +=
8/8/2019 Euler Cauchy Equation
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Example
Solution
Find the G.S. of the ODE
mxy =
0}137)1({ =++m
xmmm
Let its solution is
So (1) will be 0137)1(
122
=++
mmm
xxmxxmmx
01372
=++ yyxyx
Muhammad NadeemSEECS-NUST
im 231 +=
So general solution
01362
=++ mm
im 232 =
( )xqcxqcxy p lnsinlncos 21 +=
( )xcxcxy ln2sinln2cos 213
+=
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