Fundamental Set of Solutions „Basis‟

Fundamental Set of Solutions

Any set of linearly independent solutions of the

homogeneous linear nth-order D.E. on an interval is said to be a fundamental set of solutions on the interval.

Definition 3.3

n

I y

n

y

y

₁

,

₂

,...,

Homogeneous

173 /364

2008_O.D.E(1)

Fundamental Set of Solutions „Basis‟

Fundamental Set of Solutions

Any set of linearly independent solutions of the

homogeneous linear nth-order D.E. on an interval is said to be a fundamental set of solutions on the interval.

Definition 3.3

n

I y

n

y

y

₁

,

₂

,...,

‘Basis’

Homogeneous

174 /364

2008_O.D.E(1)

Fundamental Set of Solutions „Basis‟

Fundamental Set of Solutions

Any set of linearly independent solutions of the

homogeneous linear nth-order D.E. on an interval is said to be a fundamental set of solutions on the interval.

Definition 3.3

n

I

Existence of a Fundamental Set

There exists a fundamental set of solutions for the homogeneous linear nth-order differential equation on an interval

Theorem 3.4

I y

n

y

y

₁

,

₂

,...,

‘Basis’

Homogeneous

175 /364

2008_O.D.E(1)

Fundamental Set of Solutions „Basis‟

Fundamental Set of Solutions

Any set of linearly independent solutions of the

homogeneous linear nth-order D.E. on an interval is said to be a fundamental set of solutions on the interval.

Definition 3.3

n

I y

n

y

y

₁

,

₂

,...,

‘Basis’

“Analogous to the fact that any vector in three dimensions can be expressed as a linear combination of the linearly independent vectors “

i, j, k

Homogeneous

176 /364

2008_O.D.E(1)

Fundamental Set of Solutions „Basis‟

Fundamental Set of Solutions

Any set of linearly independent solutions of the

homogeneous linear nth-order D.E. on an interval is said to be a fundamental set of solutions on the interval.

Definition 3.3

n

I y

n

y

y

₁

,

₂

,...,

‘Basis’

“Analogous to the fact that any vector in three dimensions can be expressed as a linear combination of the linearly independent vectors “

i, j, k

k

j

a i b

c r

Homogeneous

177 /364

2008_O.D.E(1)

Fundamental Set of Solutions „Basis‟

Fundamental Set of Solutions

Any set of linearly independent solutions of the

homogeneous linear nth-order D.E. on an interval is said to be a fundamental set of solutions on the interval.

Definition 3.3

n

I y

n

y

y

₁

,

₂

,...,

‘Basis’

“Analogous to the fact that any vector in three dimensions can be expressed as a linear combination of the linearly independent vectors “

i, j, k

k

j

a i b

c r

k j

i

r  a  b  c

Homogeneous

178 /364

2008_O.D.E(1)

Fundamental Set of Solutions „Basis‟

Fundamental Set of Solutions

Any set of linearly independent solutions of the

homogeneous linear nth-order D.E. on an interval is said to be a fundamental set of solutions on the interval.

Definition 3.3

n

I y

n

y

y

₁

,

₂

,...,

‘Basis’

“Analogous to the fact that any vector in three dimensions can be expressed as a linear combination of the linearly independent vectors “

i, j, k

k

j

a i b

c r

k j

i

r  a  b  c

linearly independent

Homogeneous

179 /364

2008_O.D.E(1)

Fundamental Set of Solutions „Basis‟

Fundamental Set of Solutions

Any set of linearly independent solutions of the

homogeneous linear nth-order D.E. on an interval is said to be a fundamental set of solutions on the interval.

Definition 3.3

n

I y

n

y

y

₁

,

₂

,...,

‘Basis’

“Analogous to the fact that any vector in three dimensions can be expressed as a linear combination of the linearly independent vectors “

i, j, k

k

j

a i b

c r

k j

i

r  a  b  c

linearly independent

Basis

Homogeneous

180 /364

2008_O.D.E(1)

Fundamental Set of Solutions „Basis‟

Fundamental Set of Solutions

Any set of linearly independent solutions of the

homogeneous linear nth-order D.E. on an interval is said to be a fundamental set of solutions on the interval.

Definition 3.3

n

I y

n

y

y

₁

,

₂

,...,

‘Basis’

“Analogous to the fact that any vector in three dimensions can be expressed as a linear combination of the linearly independent vectors “

i, j, k

k

j

a i b

c r

k j

i

r  a  b  c

linearly independent

Basis

Homogeneous

k

j

i

181 /364

2008_O.D.E(1)

Fundamental Set of Solutions „Basis‟

Fundamental Set of Solutions

Any set of linearly independent solutions of the

homogeneous linear nth-order D.E. on an interval is said to be a fundamental set of solutions on the interval.

Definition 3.3

n

I y

n

y

y

₁

,

₂

,...,

‘Basis’

“Analogous to the fact that any vector in three dimensions can be expressed as a linear combination of the linearly independent vectors “

i, j, k

k

j

a i b

c r

k j

i

r  a  b  c

linearly independent

Basis

Homogeneous

k

j

i jˆ

iˆ

_/364¹⁸²

2008_O.D.E(1)

Fundamental Set of Solutions „Basis‟

Fundamental Set of Solutions

Any set of linearly independent solutions of the

homogeneous linear nth-order D.E. on an interval is said to be a fundamental set of solutions on the interval.

Definition 3.3

n

I y

n

y

y

₁

,

₂

,...,

‘Basis’

“Analogous to the fact that any vector in three dimensions can be expressed as a linear combination of the linearly independent vectors “

i, j, k

k

j

a i b

c r

k j

i

r  a  b  c

linearly independent

Basis

Homogeneous

k

j

aˆ i

bˆ

c r jˆ

iˆ

_/364¹⁸³

2008_O.D.E(1)

Fundamental Set of Solutions „Basis‟

Fundamental Set of Solutions

Any set of linearly independent solutions of the

homogeneous linear nth-order D.E. on an interval is said to be a fundamental set of solutions on the interval.

Definition 3.3

n

I y

n

y

y

₁

,

₂

,...,

‘Basis’

“Analogous to the fact that any vector in three dimensions can be expressed as a linear combination of the linearly independent vectors “

i, j, k

k

j

a i b

c r

k j

i

r  a  b  c

linearly independent

Basis

Homogeneous

k

j

aˆ i

bˆ

c r jˆ

iˆ

k j

i

r  a ˆ ˆ  b ˆ ˆ  c

184 /364

2008_O.D.E(1)

Fundamental Set of Solutions „Basis‟

Fundamental Set of Solutions

Any set of linearly independent solutions of the

homogeneous linear nth-order D.E. on an interval is said to be a fundamental set of solutions on the interval.

Definition 3.3

n

I y

n

y

y

₁

,

₂

,...,

‘Basis’

“Analogous to the fact that any vector in three dimensions can be expressed as a linear combination of the linearly independent vectors “

i, j, k

k

j

a i b

c r

k j

i

r  a  b  c

linearly independent

Basis

Homogeneous

k

j

aˆ i

bˆ

c r jˆ

iˆ

k j

i

r  a ˆ ˆ  b ˆ ˆ  c

linearly independent

185 /364

2008_O.D.E(1)

Fundamental Set of Solutions „Basis‟

Fundamental Set of Solutions

Any set of linearly independent solutions of the

homogeneous linear nth-order D.E. on an interval is said to be a fundamental set of solutions on the interval.

Definition 3.3

n

I y

n

y

y

₁

,

₂

,...,

‘Basis’

“Analogous to the fact that any vector in three dimensions can be expressed as a linear combination of the linearly independent vectors “

i, j, k

k

j

a i b

c r

k j

i

r  a  b  c

linearly independent

Basis

Homogeneous

k

j

aˆ i

bˆ

c r jˆ

iˆ

k j

i

r  a ˆ ˆ  b ˆ ˆ  c

linearly independent

New Basis

186 /364

2008_O.D.E(1)

Fundamental Set of Solutions „Basis‟

Fundamental Set of Solutions

Any set of linearly independent solutions of the

homogeneous linear nth-order D.E. on an interval is said to be a fundamental set of solutions on the interval.

Definition 3.3

n

I y

n

y

y

₁

,

₂

,...,

‘Basis’

“Analogous to the fact that any vector in three dimensions can be expressed as a linear combination of the linearly independent vectors “

i, j, k

k

j

a i b

c r

k j

i

r  a  b  c

linearly independent

Basis

Homogeneous

187 /364

2008_O.D.E(1)

Fundamental Set of Solutions „Basis‟

Fundamental Set of Solutions

Any set of linearly independent solutions of the

homogeneous linear nth-order D.E. on an interval is said to be a fundamental set of solutions on the interval.

Definition 3.3

n

I y

n

y

y

₁

,

₂

,...,

‘Basis’

“Analogous to the fact that any vector in three dimensions can be expressed as a linear combination of the linearly independent vectors “

i, j, k

k

j

a i b

c r

k j

i

r  a  b  c

linearly independent

Basis

r

j

i k

Homogeneous

188 /364

2008_O.D.E(1)

Fundamental Set of Solutions „Basis‟

Fundamental Set of Solutions

Any set of linearly independent solutions of the

homogeneous linear nth-order D.E. on an interval is said to be a fundamental set of solutions on the interval.

Definition 3.3

n

I y

n

y

y

₁

,

₂

,...,

‘Basis’

“Analogous to the fact that any vector in three dimensions can be expressed as a linear combination of the linearly independent vectors “

i, j, k

k

j

a i b

c r

k j

i

r  a  b  c

linearly independent

Basis

j i k ˆ   r

j

i k

Homogeneous

189 /364

2008_O.D.E(1)

Fundamental Set of Solutions „Basis‟

Fundamental Set of Solutions

Any set of linearly independent solutions of the

homogeneous linear nth-order D.E. on an interval is said to be a fundamental set of solutions on the interval.

Definition 3.3

n

I y

n

y

y

₁

,

₂

,...,

‘Basis’

“Analogous to the fact that any vector in three dimensions can be expressed as a linear combination of the linearly independent vectors “

i, j, k

k

j

a i b

c r

k j

i

r  a  b  c

linearly independent

Basis

  ^{i , j , k ˆ}

can be another new basis?

j i k ˆ   r

j

i

k r  a i  b j  c ˆk ˆ ??

Homogeneous

a b

190 /364

2008_O.D.E(1)

Fundamental Set of Solutions „Basis‟

Fundamental Set of Solutions

Any set of linearly independent solutions of the

homogeneous linear nth-order D.E. on an interval is said to be a fundamental set of solutions on the interval.

Definition 3.3

n

I y

n

y

y

₁

,

₂

,...,

‘Basis’

“Analogous to the fact that any vector in three dimensions can be expressed as a linear combination of the linearly independent vectors “

i, j, k

k

j

a i b

c r

k j

i

r  a  b  c

linearly independent

Basis

  ^{i , j , k ˆ}

can be another new basis?

j i k ˆ   r

j

i

k r  a i  b j  c ˆk ˆ ??

Homogeneous

a b

Linearly dependent

191 /364

2008_O.D.E(1)

Fundamental Set of Solutions „Basis‟

Fundamental Set of Solutions

Any set of linearly independent solutions of the

homogeneous linear nth-order D.E. on an interval is said to be a fundamental set of solutions on the interval.

Definition 3.3

n

I y

n

y

y

₁

,

₂

,...,

‘Basis’

“Analogous to the fact that any vector in three dimensions can be expressed as a linear combination of the linearly independent vectors “

i, j, k

k

j

a i b

c r

k j

i

r  a  b  c

linearly independent

Basis

  ^{i , j , k ˆ}

can be another new basis?

j i k ˆ   r

j

i

k r  a i  b j  c ˆk ˆ ??

Homogeneous

a b

Linearly dependent

New Basis

192 /364

2008_O.D.E(1)

Fundamental Set of Solutions „Basis‟

Fundamental Set of Solutions

Any set of linearly independent solutions of the

homogeneous linear nth-order D.E. on an interval is said to be a fundamental set of solutions on the interval.

Definition 3.3

n

I y

n

y

y

₁

,

₂

,...,

‘Basis’

Homogeneous

193 /364

2008_O.D.E(1)

Fundamental Set of Solutions „Basis‟

Fundamental Set of Solutions

Any set of linearly independent solutions of the

homogeneous linear nth-order D.E. on an interval is said to be a fundamental set of solutions on the interval.

Definition 3.3

n

I y

n

y

y

₁

,

₂

,...,

‘Basis’

“Any solution of an nth-order homogeneous linear differential equation on an interval can be expresses as a linear combination of n linearly independent solutions on .

I I

) ( )

( )

(

₂

1

x y x y x

y

_n

Homogeneous

194 /364

2008_O.D.E(1)

Fundamental Set of Solutions „Basis‟

Fundamental Set of Solutions

Any set of linearly independent solutions of the

homogeneous linear nth-order D.E. on an interval is said to be a fundamental set of solutions on the interval.

Definition 3.3

n

I y

n

y

y

₁

,

₂

,...,

‘Basis’

“Any solution of an nth-order homogeneous linear differential equation on an interval can be expresses as a linear combination of n linearly independent solutions on .

I I

) ( )

( )

(

₂

1

x y x y x

y

_n

Homogeneous

Basis Solutions (Linearly Independent)

195 /364

2008_O.D.E(1)

Fundamental Set of Solutions „Basis‟

Fundamental Set of Solutions

Any set of linearly independent solutions of the

homogeneous linear nth-order D.E. on an interval is said to be a fundamental set of solutions on the interval.

Definition 3.3

n

I y

n

y

y

₁

,

₂

,...,

‘Basis’

“Any solution of an nth-order homogeneous linear differential equation on an interval can be expresses as a linear combination of n linearly independent solutions on .

I I

) ( )

( )

(

₂

1

x y x y x

y

_n

Homogeneous

Basis Solutions (Linearly Independent)

Linear

Combination

c

n

c c

y 

₁



₂

  

196 /364

2008_O.D.E(1)

t F

kz z

c z

m      ₀ cos 

homogeneous solution m z   c z   kz  0

문서에서 Engineering Mathematics2 (페이지 173-197)