Cauchy Euler Equation, Exams of Mathematics

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MATH 2310 / 2320 Differential EquationsDifferential Equations^ 6. Cauchy-Euler Equations

Dr. Faried Hasbullah | Sem 1 - 2017/

Zill, D.G. & Wright, W.S., (2017)

Section 4.

Cauchy-Euler Equation: Definition
^

A linear differential equation of the form

ᡦ)^

+^

−^1

−^1

ᡦ−

1)^

+^

+^

ᡓ^2

)^

ᡦ^

ᡓ^

ᡶ^

ᡷ^

ᡓ^

ᡓ^

ᡷ^

ᡙ^

where the coefficients

,^ ᡦ

−^1

are constants, is 0

called a

Cauchy-Euler equation

ᡦ)

Eg.

Method of solution
^

We try a solution of the form

ᡷ

=ᡶ

ᡥ, where

ᡥ

is to be

determined.
^

When we substitute

ᡷ

=ᡶ

ᡥ, the second-order equation

becomes

ᡓᡶ

2 ᡥ

(ᡥ

−1)

ᡥᡶ -2+

ᡔᡶ

ᡥ

ᡥᡶ -1+

ᡕᡶ

ᡥ=

ᡓᡥ

(ᡥ

−1)

ᡥᡶ +ᡔ

ᡥ

ᡥᡶ +ᡕ

ᡥᡶ

ᡓᡥ

ᡓᡥ

ᡔ

ᡥ

ᡕ

ᡶ

ᡥ

ᡓᡶ

ᡥ

ᡥ

ᡥᡶ

ᡔᡶ

ᡥ

ᡥᡶ

ᡕᡶ

ᡥ

ᡓᡥ

(ᡥ

−1)

ᡥᡶ +ᡔ

ᡥ

ᡥᡶ +ᡕ

ᡥᡶ

(ᡓ

ᡥ

2 −

ᡓᡥ

+ᡔ

ᡥ

+ᡕ

)ᡶ

ᡥ=

ᡓᡥ

2 +(

ᡔ−

ᡓ)

ᡥ

+ᡕ

=

• (3)

The last equation is the characteristic equation of the differential equation (2).

Caution!This characteristic equation is only forsecond order Cauchy-Euler equations!You’ve to derive the characteristic equationfor third order and above.

Solution of Cauchy-Euler Equation
^

Case I:

Let

and 1

denote 2

real and distinct

roots of

(3), then general solution of (2) is

ᠩ^1

ᡥ^2

^

Case II:

Let

and 1

be 2

real and equal

roots of (3),

where

, then general solution of (2) is 1

ᡷ^

ᠩ^

ᠩ^

2

1

ᡷ^

=^

ᠩ^1

ᡥ^1

ln

^

Case III:

Conjugate complex roots

If the roots (3)

+^

and

−^

where

,^ ‐

> 0, then general solution is ᡷ^

=^

ᡶ^

[ᠩ

cos( 1

‐ln

sin( 2

‐ln

ᡶ)]

Example 2 Solve 4

′^ +

(^

(^

m m

m

m

c ma b

am

c b a

0 1 4 4

0 1 4 8

4

0 1 , 8 , (^4222)

=

=

−

=

−

=

=

x xc

xc y

m m m

a

ac b b

m

ln 1 2

) (^4) ( 2

) 1 )( (^4) ( 4 4 4

2

4 1 2 2 1 2 (^21) 1

2

(^2) , 1

2

(^2) , 1

−

−

=

−=

=

− ± − =

− ± − =

1

1 2

2

1

2

ln

y^

c x^

c x

x

−^

−

=^

Example 3 Solve 4

ᡷ^

(^

(^

4, 2 2 2

0,^

17

0

4

0

4

17

0

4

4

17

0

a^

b^

c

am

b^

a m

c

m^

m

m^

m =^

=^

=

+^

−^

+^

=

+^

−^

+^

=

−^

+^

= (^

)^

(^

2

1,

2

1,2 1,

1/

1

2

4 2 4

4

4(4)(17) 2(4)

4

16

1

2

8

2 cos 2ln

sin 2ln

b^

b^

ac

m^

a

m

i

m^

i

y^

x^

c^

x^

c^

x

−^

±^

−

=

±^

−^

−

=

± =^

=^

±

=^

^



^



(^

)^

(^

)

1/

1

2

cos 2ln

sin 2ln

y^

x^

c^

x^

c^

x

=^

^



^



Example 5 Solve

0 8

7

5

2 2 2

3 3 3

=

+^

y

dyx dx

dx

y d x

dx

y d x

(^

)^

(^

)

2 1

2

3

cos 2 ln

sin 2ln

y^

c x

c^

x^

c^

x

− =^

+^

Homework

Ex 4.7: 13, 15, 23, 37 Textbook:Zill, D.G & Cullen, M.R, (2009), Differential Equationswith Boundary-Value Problems.

th ( or any edition)

orZill, D.G. & Wright,

W.S, (2017), Differential Equations

with Boundary-Value Problems.

th ( edition | Metric version)