The Method of Frobenius Bernd Schr oder logo1 Bernd Schr oder - - PowerPoint PPT Presentation

logo1 Overview An Example Double Check

The Method of Frobenius

Bernd Schr¨

• der

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

What is the Method of Frobenius?

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

What is the Method of Frobenius?

• 1. The method of Frobenius works for differential equations
• f the form y′′ +P(x)y′ +Q(x)y = 0 in which P or Q is not

analytic at the point of expansion x0.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

What is the Method of Frobenius?

• 1. The method of Frobenius works for differential equations
• f the form y′′ +P(x)y′ +Q(x)y = 0 in which P or Q is not

analytic at the point of expansion x0.

• 2. But P and Q cannot be arbitrary: (x−x0)P(x) and

(x−x0)2Q(x) must be analytic at x0.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

What is the Method of Frobenius?

• 1. The method of Frobenius works for differential equations
• f the form y′′ +P(x)y′ +Q(x)y = 0 in which P or Q is not

analytic at the point of expansion x0.

• 2. But P and Q cannot be arbitrary: (x−x0)P(x) and

(x−x0)2Q(x) must be analytic at x0.

• 3. Instead of a series solution y =

∞

∑

n=0

cn(x−x0)n, we obtain a solution of the form y =

∞

∑

n=0

cn(x−x0)n+r.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

What is the Method of Frobenius?

• 1. The method of Frobenius works for differential equations
• f the form y′′ +P(x)y′ +Q(x)y = 0 in which P or Q is not

analytic at the point of expansion x0.

• 2. But P and Q cannot be arbitrary: (x−x0)P(x) and

(x−x0)2Q(x) must be analytic at x0.

• 3. Instead of a series solution y =

∞

∑

n=0

cn(x−x0)n, we obtain a solution of the form y =

∞

∑

n=0

cn(x−x0)n+r.

• 4. The method of Frobenius is guaranteed to produce one

solution, but it may not produce two linearly independent solutions.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

What is the Method of Frobenius?

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

What is the Method of Frobenius?

• 5. As for series solutions, we substitute the series and its

derivatives into the equation to obtain an equation for r and a set of equations for the cn.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

What is the Method of Frobenius?

• 5. As for series solutions, we substitute the series and its

derivatives into the equation to obtain an equation for r and a set of equations for the cn.

• 6. These equations will allow us to compute r and the cn.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

What is the Method of Frobenius?

• 5. As for series solutions, we substitute the series and its

derivatives into the equation to obtain an equation for r and a set of equations for the cn.

• 6. These equations will allow us to compute r and the cn.
• 7. For each value of r (typically there are two), we can

compute the solution just like for series.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

What is the Method of Frobenius?

• 5. As for series solutions, we substitute the series and its

derivatives into the equation to obtain an equation for r and a set of equations for the cn.

• 6. These equations will allow us to compute r and the cn.
• 7. For each value of r (typically there are two), we can

compute the solution just like for series. That’s it.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

Solve the Differential Equation 9x2y′′ +3x2y′ +2y = 0

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

Solve the Differential Equation 9x2y′′ +3x2y′ +2y = 0

9x2y′′ +3x2y′ +2y =

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

Solve the Differential Equation 9x2y′′ +3x2y′ +2y = 0

9x2y′′ +3x2y′ +2y = 9x2

• ∞

∑

n=0

cnxn+r ′′ +3x2

• ∞

∑

n=0

cnxn+r ′ +2

• ∞

∑

n=0

cnxn+r

• =

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

Solve the Differential Equation 9x2y′′ +3x2y′ +2y = 0

9x2y′′ +3x2y′ +2y = 9x2

• ∞

∑

n=0

cnxn+r ′′ +3x2

• ∞

∑

n=0

cnxn+r ′ +2

∞

∑

n=0

cnxn+r =

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

Solve the Differential Equation 9x2y′′ +3x2y′ +2y = 0

9x2y′′ +3x2y′ +2y = 9x2

• ∞

∑

n=0

cnxn+r ′′ +3x2

∞

∑

n=0

cn(n+r)xn+r−1 +2

∞

∑

n=0

cnxn+r =

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

Solve the Differential Equation 9x2y′′ +3x2y′ +2y = 0

9x2y′′ +3x2y′ +2y = 9x2

∞

∑

n=0

cn(n+r)(n+r −1)xn+r−2 +3x2

∞

∑

n=0

cn(n+r)xn+r−1 +2

∞

∑

n=0

cnxn+r =

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

Solve the Differential Equation 9x2y′′ +3x2y′ +2y = 0

9x2y′′ +3x2y′ +2y = 9x2

∞

∑

n=0

cn(n+r)(n+r −1)xn+r−2 +3x2

∞

∑

n=0

cn(n+r)xn+r−1 +2

∞

∑

n=0

cnxn+r =

∞

∑

n=0

9(n+r)(n+r −1)cnxn+r +

∞

∑

n=0

3(n+r)cnxn+r+1 +

∞

∑

n=0

2cnxn+r =

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

Solve the Differential Equation 9x2y′′ +3x2y′ +2y = 0

9x2y′′ +3x2y′ +2y = 9x2

∞

∑

n=0

cn(n+r)(n+r −1)xn+r−2 +3x2

∞

∑

n=0

cn(n+r)xn+r−1 +2

∞

∑

n=0

cnxn+r =

∞

∑

n=0

9(n+r)(n+r −1)cnxn+r +

∞

∑

n=0

3(n+r)cnxn+r+1 +

∞

∑

n=0

2cnxn+r =

∞

∑

k=0

9(k +r)(k +r −1)ckxk+r +

∞

∑

k=1

3(k +r −1)ck−1xk+r +

∞

∑

k=0

2ckxk+r =

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

Solve the Differential Equation 9x2y′′ +3x2y′ +2y = 0

9x2y′′ +3x2y′ +2y = 9x2

∞

∑

n=0

cn(n+r)(n+r −1)xn+r−2 +3x2

∞

∑

n=0

cn(n+r)xn+r−1 +2

∞

∑

n=0

cnxn+r =

∞

∑

n=0

9(n+r)(n+r −1)cnxn+r +

∞

∑

n=0

3(n+r)cnxn+r+1 +

∞

∑

n=0

2cnxn+r =

∞

∑

k=0

9(k +r)(k +r −1)ckxk+r +

∞

∑

k=1

3(k +r −1)ck−1xk+r +

∞

∑

k=0

2ckxk+r =

• 9r(r−1)+2
• c0xr+

∞

∑

k=1

• 9(k+r)(k+r−1)ck +3(k+r−1)ck−1+2ck
• xk+r

=

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

Solve the Differential Equation 9x2y′′ +3x2y′ +2y = 0

Indicial equation:

• 9r(r −1)+2
• c0

=

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

Solve the Differential Equation 9x2y′′ +3x2y′ +2y = 0

Indicial equation:

• 9r(r −1)+2
• c0

= 9r(r −1)+2 =

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

Solve the Differential Equation 9x2y′′ +3x2y′ +2y = 0

Indicial equation:

• 9r(r −1)+2
• c0

= 9r(r −1)+2 = 9r2 −9r +2 =

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

Solve the Differential Equation 9x2y′′ +3x2y′ +2y = 0

Indicial equation:

• 9r(r −1)+2
• c0

= 9r(r −1)+2 = 9r2 −9r +2 = r1,2 = 9± √ 81−72 18

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

Solve the Differential Equation 9x2y′′ +3x2y′ +2y = 0

Indicial equation:

• 9r(r −1)+2
• c0

= 9r(r −1)+2 = 9r2 −9r +2 = r1,2 = 9± √ 81−72 18 = 9±3 18

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

Solve the Differential Equation 9x2y′′ +3x2y′ +2y = 0

Indicial equation:

• 9r(r −1)+2
• c0

= 9r(r −1)+2 = 9r2 −9r +2 = r1,2 = 9± √ 81−72 18 = 9±3 18 = 2 3

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

Solve the Differential Equation 9x2y′′ +3x2y′ +2y = 0

Indicial equation:

• 9r(r −1)+2
• c0

= 9r(r −1)+2 = 9r2 −9r +2 = r1,2 = 9± √ 81−72 18 = 9±3 18 = 2 3, 1 3

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

Solve the Differential Equation 9x2y′′ +3x2y′ +2y = 0

Recurrence Relation.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

Solve the Differential Equation 9x2y′′ +3x2y′ +2y = 0

Recurrence Relation. 9(k +r)(k +r −1)ck +3(k +r −1)ck−1 +2ck =

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

Solve the Differential Equation 9x2y′′ +3x2y′ +2y = 0

Recurrence Relation. 9(k +r)(k +r −1)ck +3(k +r −1)ck−1 +2ck =

• 9(k +r)(k +r −1)+2
• ck +3(k +r −1)ck−1

=

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

Solve the Differential Equation 9x2y′′ +3x2y′ +2y = 0

Recurrence Relation. 9(k +r)(k +r −1)ck +3(k +r −1)ck−1 +2ck =

• 9(k +r)(k +r −1)+2
• ck +3(k +r −1)ck−1

=

• 9(k +r)(k +r −1)+2
• ck

= −3(k +r −1)ck−1

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

Solve the Differential Equation 9x2y′′ +3x2y′ +2y = 0

Recurrence Relation. 9(k +r)(k +r −1)ck +3(k +r −1)ck−1 +2ck =

• 9(k +r)(k +r −1)+2
• ck +3(k +r −1)ck−1

=

• 9(k +r)(k +r −1)+2
• ck

= −3(k +r −1)ck−1 ck = 3−3k −3r 9(k +r)(k +r −1)+2ck−1

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

Solve the Differential Equation 9x2y′′ +3x2y′ +2y = 0

r = 2 3

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

Solve the Differential Equation 9x2y′′ +3x2y′ +2y = 0

r = 2 3 , c0 = 1

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

Solve the Differential Equation 9x2y′′ +3x2y′ +2y = 0

r = 2 3 , c0 = 1 ck = 3−3k −3r 9(k +r)(k +r −1)+2ck−1

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

Solve the Differential Equation 9x2y′′ +3x2y′ +2y = 0

r = 2 3 , c0 = 1 ck = 3−3k −3r 9(k +r)(k +r −1)+2ck−1 = 3−3k −3· 2

3

3

• k + 2

3

• 3
• k + 2

3 −1

• +2ck−1

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

Solve the Differential Equation 9x2y′′ +3x2y′ +2y = 0

r = 2 3 , c0 = 1 ck = 3−3k −3r 9(k +r)(k +r −1)+2ck−1 = 3−3k −3· 2

3

3

• k + 2

3

• 3
• k + 2

3 −1

• +2ck−1

= 1−3k (3k +2)(3k −1)+2ck−1

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

Solve the Differential Equation 9x2y′′ +3x2y′ +2y = 0

r = 2 3 , c0 = 1 ck = 3−3k −3r 9(k +r)(k +r −1)+2ck−1 = 3−3k −3· 2

3

3

• k + 2

3

• 3
• k + 2

3 −1

• +2ck−1

= 1−3k (3k +2)(3k −1)+2ck−1 = 1−3k 9k2 +3kck−1

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

Solve the Differential Equation 9x2y′′ +3x2y′ +2y = 0

r = 2 3 , c0 = 1 ck = 3−3k −3r 9(k +r)(k +r −1)+2ck−1 = 3−3k −3· 2

3

3

• k + 2

3

• 3
• k + 2

3 −1

• +2ck−1

= 1−3k (3k +2)(3k −1)+2ck−1 = 1−3k 9k2 +3kck−1 c1 = 1−3·1 9·12 +3·1c1−1

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

Solve the Differential Equation 9x2y′′ +3x2y′ +2y = 0

r = 2 3 , c0 = 1 ck = 3−3k −3r 9(k +r)(k +r −1)+2ck−1 = 3−3k −3· 2

3

3

• k + 2

3

• 3
• k + 2

3 −1

• +2ck−1

= 1−3k (3k +2)(3k −1)+2ck−1 = 1−3k 9k2 +3kck−1 c1 = 1−3·1 9·12 +3·1c1−1 = − 2 12 ·1

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

Solve the Differential Equation 9x2y′′ +3x2y′ +2y = 0

r = 2 3 , c0 = 1 ck = 3−3k −3r 9(k +r)(k +r −1)+2ck−1 = 3−3k −3· 2

3

3

• k + 2

3

• 3
• k + 2

3 −1

• +2ck−1

= 1−3k (3k +2)(3k −1)+2ck−1 = 1−3k 9k2 +3kck−1 c1 = 1−3·1 9·12 +3·1c1−1 = − 2 12 ·1 = −1 6

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

Solve the Differential Equation 9x2y′′ +3x2y′ +2y = 0

r = 2 3 , c0 = 1 ck = 3−3k −3r 9(k +r)(k +r −1)+2ck−1 = 3−3k −3· 2

3

3

• k + 2

3

• 3
• k + 2

3 −1

• +2ck−1

= 1−3k (3k +2)(3k −1)+2ck−1 = 1−3k 9k2 +3kck−1 c1 = 1−3·1 9·12 +3·1c1−1 = − 2 12 ·1 = −1 6 c2 = 1−3·2 9·22 +3·2c2−1

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

Solve the Differential Equation 9x2y′′ +3x2y′ +2y = 0

r = 2 3 , c0 = 1 ck = 3−3k −3r 9(k +r)(k +r −1)+2ck−1 = 3−3k −3· 2

3

3

• k + 2

3

• 3
• k + 2

3 −1

• +2ck−1

= 1−3k (3k +2)(3k −1)+2ck−1 = 1−3k 9k2 +3kck−1 c1 = 1−3·1 9·12 +3·1c1−1 = − 2 12 ·1 = −1 6 c2 = 1−3·2 9·22 +3·2c2−1 = − 5 42

• −1

6

• Bernd Schr¨
• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

Solve the Differential Equation 9x2y′′ +3x2y′ +2y = 0

r = 2 3 , c0 = 1 ck = 3−3k −3r 9(k +r)(k +r −1)+2ck−1 = 3−3k −3· 2

3

3

• k + 2

3

• 3
• k + 2

3 −1

• +2ck−1

= 1−3k (3k +2)(3k −1)+2ck−1 = 1−3k 9k2 +3kck−1 c1 = 1−3·1 9·12 +3·1c1−1 = − 2 12 ·1 = −1 6 c2 = 1−3·2 9·22 +3·2c2−1 = − 5 42

• −1

6

• =

5 252

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

Solve the Differential Equation 9x2y′′ +3x2y′ +2y = 0

r = 2 3 , c0 = 1 ck = 3−3k −3r 9(k +r)(k +r −1)+2ck−1 = 3−3k −3· 2

3

3

• k + 2

3

• 3
• k + 2

3 −1

• +2ck−1

= 1−3k (3k +2)(3k −1)+2ck−1 = 1−3k 9k2 +3kck−1 c1 = 1−3·1 9·12 +3·1c1−1 = − 2 12 ·1 = −1 6 c2 = 1−3·2 9·22 +3·2c2−1 = − 5 42

• −1

6

• =

5 252 c3 = 1−3·3 9·32 +3·3c3−1

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

Solve the Differential Equation 9x2y′′ +3x2y′ +2y = 0

r = 2 3 , c0 = 1 ck = 3−3k −3r 9(k +r)(k +r −1)+2ck−1 = 3−3k −3· 2

3

3

• k + 2

3

• 3
• k + 2

3 −1

• +2ck−1

= 1−3k (3k +2)(3k −1)+2ck−1 = 1−3k 9k2 +3kck−1 c1 = 1−3·1 9·12 +3·1c1−1 = − 2 12 ·1 = −1 6 c2 = 1−3·2 9·22 +3·2c2−1 = − 5 42

• −1

6

• =

5 252 c3 = 1−3·3 9·32 +3·3c3−1 = − 8 90 5 252

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

Solve the Differential Equation 9x2y′′ +3x2y′ +2y = 0

r = 2 3 , c0 = 1 ck = 3−3k −3r 9(k +r)(k +r −1)+2ck−1 = 3−3k −3· 2

3

3

• k + 2

3

• 3
• k + 2

3 −1

• +2ck−1

= 1−3k (3k +2)(3k −1)+2ck−1 = 1−3k 9k2 +3kck−1 c1 = 1−3·1 9·12 +3·1c1−1 = − 2 12 ·1 = −1 6 c2 = 1−3·2 9·22 +3·2c2−1 = − 5 42

• −1

6

• =

5 252 c3 = 1−3·3 9·32 +3·3c3−1 = − 8 90 5 252 = − 1 567

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

Solve the Differential Equation 9x2y′′ +3x2y′ +2y = 0

r = 2 3 c4 = 1−3·4 9·42 +3·4c4−1

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

Solve the Differential Equation 9x2y′′ +3x2y′ +2y = 0

r = 2 3 c4 = 1−3·4 9·42 +3·4c4−1 = − 11 156

• − 1

567

• Bernd Schr¨
• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

Solve the Differential Equation 9x2y′′ +3x2y′ +2y = 0

r = 2 3 c4 = 1−3·4 9·42 +3·4c4−1 = − 11 156

• − 1

567

• =

11 88452

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

Solve the Differential Equation 9x2y′′ +3x2y′ +2y = 0

r = 2 3 c4 = 1−3·4 9·42 +3·4c4−1 = − 11 156

• − 1

567

• =

11 88452 y1 = x

2 3 − 1

6x

2 3 +1 + 5

252x

2 3 +2 − 1

567x

2 3 +3 +

11 88452x

2 3 +4 Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

Solve the Differential Equation 9x2y′′ +3x2y′ +2y = 0

r = 2 3 c4 = 1−3·4 9·42 +3·4c4−1 = − 11 156

• − 1

567

• =

11 88452 y1 = x

2 3 − 1

6x

2 3 +1 + 5

252x

2 3 +2 − 1

567x

2 3 +3 +

11 88452x

2 3 +4

= x

2 3 − 1

6x

5 3 + 5

252x

8 3 − 1

567x

11 3 +

11 88452x

14 3 Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

Solve the Differential Equation 9x2y′′ +3x2y′ +2y = 0

r = 1 3

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

Solve the Differential Equation 9x2y′′ +3x2y′ +2y = 0

r = 1 3 , c0 = 1

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

Solve the Differential Equation 9x2y′′ +3x2y′ +2y = 0

r = 1 3 , c0 = 1 ck = 3−3k −3r 9(k +r)(k +r −1)+2ck−1

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

Solve the Differential Equation 9x2y′′ +3x2y′ +2y = 0

r = 1 3 , c0 = 1 ck = 3−3k −3r 9(k +r)(k +r −1)+2ck−1 = 3−3k −3· 1

3

3

• k + 1

3

• 3
• k + 1

3 −1

• +2ck−1

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

Solve the Differential Equation 9x2y′′ +3x2y′ +2y = 0

r = 1 3 , c0 = 1 ck = 3−3k −3r 9(k +r)(k +r −1)+2ck−1 = 3−3k −3· 1

3

3

• k + 1

3

• 3
• k + 1

3 −1

• +2ck−1

= 2−3k (3k +1)(3k −2)+2ck−1

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

Solve the Differential Equation 9x2y′′ +3x2y′ +2y = 0

r = 1 3 , c0 = 1 ck = 3−3k −3r 9(k +r)(k +r −1)+2ck−1 = 3−3k −3· 1

3

3

• k + 1

3

• 3
• k + 1

3 −1

• +2ck−1

= 2−3k (3k +1)(3k −2)+2ck−1 = 2−3k 9k2 −3kck−1

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

Solve the Differential Equation 9x2y′′ +3x2y′ +2y = 0

r = 1 3 , c0 = 1 ck = 3−3k −3r 9(k +r)(k +r −1)+2ck−1 = 3−3k −3· 1

3

3

• k + 1

3

• 3
• k + 1

3 −1

• +2ck−1

= 2−3k (3k +1)(3k −2)+2ck−1 = 2−3k 9k2 −3kck−1 c1 = 2−3·1 9·12 −3·1c1−1

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

Solve the Differential Equation 9x2y′′ +3x2y′ +2y = 0

r = 1 3 , c0 = 1 ck = 3−3k −3r 9(k +r)(k +r −1)+2ck−1 = 3−3k −3· 1

3

3

• k + 1

3

• 3
• k + 1

3 −1

• +2ck−1

= 2−3k (3k +1)(3k −2)+2ck−1 = 2−3k 9k2 −3kck−1 c1 = 2−3·1 9·12 −3·1c1−1 = −1 6

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

Solve the Differential Equation 9x2y′′ +3x2y′ +2y = 0

r = 1 3 , c0 = 1 ck = 3−3k −3r 9(k +r)(k +r −1)+2ck−1 = 3−3k −3· 1

3

3

• k + 1

3

• 3
• k + 1

3 −1

• +2ck−1

= 2−3k (3k +1)(3k −2)+2ck−1 = 2−3k 9k2 −3kck−1 c1 = 2−3·1 9·12 −3·1c1−1 = −1 6 c2 = 2−3·2 9·22 −3·2c2−1

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

Solve the Differential Equation 9x2y′′ +3x2y′ +2y = 0

r = 1 3 , c0 = 1 ck = 3−3k −3r 9(k +r)(k +r −1)+2ck−1 = 3−3k −3· 1

3

3

• k + 1

3

• 3
• k + 1

3 −1

• +2ck−1

= 2−3k (3k +1)(3k −2)+2ck−1 = 2−3k 9k2 −3kck−1 c1 = 2−3·1 9·12 −3·1c1−1 = −1 6 c2 = 2−3·2 9·22 −3·2c2−1 = − 4 30

• −1

6

• Bernd Schr¨
• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

Solve the Differential Equation 9x2y′′ +3x2y′ +2y = 0

r = 1 3 , c0 = 1 ck = 3−3k −3r 9(k +r)(k +r −1)+2ck−1 = 3−3k −3· 1

3

3

• k + 1

3

• 3
• k + 1

3 −1

• +2ck−1

= 2−3k (3k +1)(3k −2)+2ck−1 = 2−3k 9k2 −3kck−1 c1 = 2−3·1 9·12 −3·1c1−1 = −1 6 c2 = 2−3·2 9·22 −3·2c2−1 = − 4 30

• −1

6

• = 1

45

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

Solve the Differential Equation 9x2y′′ +3x2y′ +2y = 0

r = 1 3 , c0 = 1 ck = 3−3k −3r 9(k +r)(k +r −1)+2ck−1 = 3−3k −3· 1

3

3

• k + 1

3

• 3
• k + 1

3 −1

• +2ck−1

= 2−3k (3k +1)(3k −2)+2ck−1 = 2−3k 9k2 −3kck−1 c1 = 2−3·1 9·12 −3·1c1−1 = −1 6 c2 = 2−3·2 9·22 −3·2c2−1 = − 4 30

• −1

6

• = 1

45 c3 = 2−3·3 9·32 −3·3c3−1

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

Solve the Differential Equation 9x2y′′ +3x2y′ +2y = 0

r = 1 3 , c0 = 1 ck = 3−3k −3r 9(k +r)(k +r −1)+2ck−1 = 3−3k −3· 1

3

3

• k + 1

3

• 3
• k + 1

3 −1

• +2ck−1

= 2−3k (3k +1)(3k −2)+2ck−1 = 2−3k 9k2 −3kck−1 c1 = 2−3·1 9·12 −3·1c1−1 = −1 6 c2 = 2−3·2 9·22 −3·2c2−1 = − 4 30

• −1

6

• = 1

45 c3 = 2−3·3 9·32 −3·3c3−1 = − 7 72 1 45

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

Solve the Differential Equation 9x2y′′ +3x2y′ +2y = 0

r = 1 3 , c0 = 1 ck = 3−3k −3r 9(k +r)(k +r −1)+2ck−1 = 3−3k −3· 1

3

3

• k + 1

3

• 3
• k + 1

3 −1

• +2ck−1

= 2−3k (3k +1)(3k −2)+2ck−1 = 2−3k 9k2 −3kck−1 c1 = 2−3·1 9·12 −3·1c1−1 = −1 6 c2 = 2−3·2 9·22 −3·2c2−1 = − 4 30

• −1

6

• = 1

45 c3 = 2−3·3 9·32 −3·3c3−1 = − 7 72 1 45 = − 7 3240

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

Solve the Differential Equation 9x2y′′ +3x2y′ +2y = 0

r = 1 3 c4 = 2−3·4 9·42 −3·4c4−1

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

Solve the Differential Equation 9x2y′′ +3x2y′ +2y = 0

r = 1 3 c4 = 2−3·4 9·42 −3·4c4−1 = − 10 132

• −

7 3240

• Bernd Schr¨
• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

Solve the Differential Equation 9x2y′′ +3x2y′ +2y = 0

r = 1 3 c4 = 2−3·4 9·42 −3·4c4−1 = − 10 132

• −

7 3240

• =

7 42768

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

Solve the Differential Equation 9x2y′′ +3x2y′ +2y = 0

r = 1 3 c4 = 2−3·4 9·42 −3·4c4−1 = − 10 132

• −

7 3240

• =

7 42768 y2 = x

1 3 − 1

6x

1 3 +1 + 1

45x

1 3+2 −

7 3240x

1 3 +3 +

7 42768x

1 3 +4 Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

Solve the Differential Equation 9x2y′′ +3x2y′ +2y = 0

r = 1 3 c4 = 2−3·4 9·42 −3·4c4−1 = − 10 132

• −

7 3240

• =

7 42768 y2 = x

1 3 − 1

6x

1 3 +1 + 1

45x

1 3+2 −

7 3240x

1 3 +3 +

7 42768x

1 3 +4

y2 = x

1 3 − 1

6x

4 3 + 1

45x

7 3 −

7 3240x

10 3 +

7 42768x

13 3 Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

Checking Solutions When Only The First Few Terms are Available

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

Checking Solutions When Only The First Few Terms are Available

As for series solutions, because the “solution” is a truncated series,we cannot expect that the differential equation is exactly satisfied.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

Checking Solutions When Only The First Few Terms are Available

As for series solutions, because the “solution” is a truncated series,we cannot expect that the differential equation is exactly satisfied.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

Checking Solutions When Only The First Few Terms are Available

As for series solutions, because the “solution” is a truncated series,we cannot expect that the differential equation is exactly satisfied.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

Checking Solutions When Only The First Few Terms are Available

As for series solutions, because the “solution” is a truncated series,we cannot expect that the differential equation is exactly satisfied.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

Checking Solutions When Only The First Few Terms are Available

As for series solutions, because the “solution” is a truncated series,we cannot expect that the differential equation is exactly satisfied.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

Checking Solutions When Only The First Few Terms are Available

As before, for a correct approximation, low order terms should cancel.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

Checking Solutions When Only The First Few Terms are Available

As before, for a correct approximation, low order terms should cancel. We went to order 14 3 for the first solution and to order 13 3 for the second solution.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius

logo1 Overview An Example Double Check

Checking Solutions When Only The First Few Terms are Available

As before, for a correct approximation, low order terms should cancel. We went to order 14 3 for the first solution and to order 13 3 for the second solution. And, of course, we use a computer algebra system.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Method of Frobenius