Answer: Find the eigenvalues and eigenfunctions.

What is a power series? Can it contain negative or fractional powers? How would you test for convergence?

Why could we use the power series method for Legendre's equation but needed the Frobenius method for Bessel's equation?

Why did we introduce twO kinds of Bessel functions, J and Y?

What is the hypergeometric equation and why did Gauss introduce it?

List the three cases of the Frobenius method, giving examples of your own.

What is the difference between an initial value problem and a boundary value problem?

What does orthogonality of functions mean and how is it used in series expansions? Give examples.

What is the Sturm-Liouville theory and its practical importance?

What do you remember about the orthogonality of the Legendre polynomials? Of Bessel functions?

What is completeness of orthogonal sets? Why is it important?

Find a basis of solutions. Try to identify the series as expansions of known functions. (Show the details of your work.) y “- 9y = 0

Find a basis of solutions. Try to identify the series as expansions of known functions. (Show the details of your work.) \((1-x)^{2} y^{\prime \prime}+(1-x) y^{\prime}-3 y=0\) Text Transcription: (1 - x)^2 y “ + (1 - x) y ‘- 3y = 0

Find a basis of solutions. Try to identify the series as expansions of known functions. (Show the details of your work.) xy “- (x + 1) y ‘+ y = 0

Find a basis of solutions. Try to identify the series as expansions of known functions. (Show the details of your work.) \(x^{2} y^{\prime \prime}-3 x y^{\prime}+4 y=0\) Text Transcription: x^2 y “ - 3xy ‘ + 4y = 0

Find a basis of solutions. Try to identify the series as expansions of known functions. (Show the details of your work.) \(y^{\prime \prime}+4 x y^{\prime}+\left(4 x^{2}+2\right) y=0\) Text Transcription: y “+ 4xy ‘+ (4x^2 + 2) y = 0

Find a basis of solutions. Try to identify the series as expansions of known functions. (Show the details of your work.) \(x^{2} y^{\prime \prime}-4 x y^{\prime}+\left(x^{2}+6\right) y=0\) Text Transcription: x^2 y “ - 4xy ‘ + (x^2 + 6) y = 0

Find a basis of solutions. Try to identify the series as expansions of known functions. (Show the details of your work.) xy “+ (2x + 1) y ‘ + (x + 1) y = 0

Find a basis of solutions. Try to identify the series as expansions of known functions. (Show the details of your work.) \(\left(x^{2}-1\right) y^{\prime \prime}-2 x y^{\prime}+2 y=0\) Text Transcription: (x^2 - 1) y “ - 2xy ‘ + 2y = 0

Find a basis of solutions. Try to identify the series as expansions of known functions. (Show the details of your work.) \(\left(x^{2}-1\right) y^{\prime \prime}+4 x y^{\prime}+2 y=0\) Text Transcription: (x^2 - 1) y “ + 4xy ‘ + 2y = 0

Find a basis of solutions. Try to identify the series as expansions of known functions. (Show the details of your work.) \(x^{2} y^{\prime \prime}+x y^{\prime}+\left(4 x^{4}-1\right) y=0\) Text Transcription: x^2 y “ + xy ‘ + (4x^4 - 1) y = 0

Find a general solution in terms of Bessel functions. (Use the indicated transformations and show the details.) \(x^{2} y^{\prime \prime}+x y^{\prime}+\left(36 x^{2}-2\right) y=0 \quad(6 x=z)\) Text Transcription: x^2 y “ + xy ‘ + (36x^2 - 2) y = 0 (6x = z)

Find a general solution in terms of Bessel functions. (Use the indicated transformations and show the details.) \(x^{2} y^{\prime \prime}+5 x y^{\prime}+\left(x^{2}-12\right) y=0 \quad\left(y=u / x^{2}\right)\) Text Transcription: x^2 y “ + 5xy ‘ + (x^2 - 12) y = 0 (y = u/x^2)

Find a general solution in terms of Bessel functions. (Use the indicated transformations and show the details.) \(x^{2} y^{\prime \prime}+x y^{\prime}+4\left(x^{4}-1\right) y=0 \quad\left(x^{2}=z\right)\) Text Transcription: x^2 y “ + xy ‘ + 4 (x^4 - 1) y = 0 (x^2 = z)

Find a general solution in terms of Bessel functions. (Use the indicated transformations and show the details.) \(4 x^{2} y^{\prime \prime}-20 x y^{\prime}+\left(4 x^{2}+35\right) y=0 \quad\left(y=x^{3} u\right)\) Text Transcription: 4x^2 y “ - 20xy ‘ + (4x^2 + 35) y = 0 (y = x^3 u)

Find a general solution in terms of Bessel functions. (Use the indicated transformations and show the details.) \(y^{\prime \prime}+k^{2} x^{2} y=0 \quad\left(y=u \sqrt{x}, \quad \frac{1}{2} k x^{2}=z\right)\) Text Transcription: y “ + k^2 x^2 y = 0 (y = u sqrt{x}, 1 / 2 kx^2 = z)

Find the eigenvalues and eigenfunctions. \(y^{\prime \prime}+\lambda y=0, \quad y(0)=0, \quad y^{\prime}(\pi)=0\) Text Transcription: y “ + lambda y = 0, y(0) = 0, y ‘ (pi) = 0

Find the eigenvalues and eigenfunctions. \(y^{\prime \prime}+\lambda y=0, \quad y(0)=y(1)\), y ‘ (0) = y ‘ (1) Text Transcription: y “ + lambda y = 0, y(0) = y(1)

Find the eigenvalues and eigenfunctions. \(\left(x y^{\prime}\right)^{\prime}+\lambda x^{-1} y=0, \quad y(1)=0, \quad y(e)=0\). (Set \(x=e^{t}\).) Text Transcription: (xy ‘) ‘ + lambda x^{-1} y = 0, y(1) = 0, y(e) = 0 x = e^t

Find the eigenvalues and eigenfunctions. \(x^{2} y^{\prime \prime}+x y^{\prime}+\left(\lambda x^{2}-1\right) y=0\), y(0) = 0, y(1) = 0 Text Transcription: x^2 y “ + xy ‘ + (lambda x^2 - 1) y = 0

Find the eigenvalues and eigenfunctions. \(y^{\prime \prime}+\lambda y=0, \quad y(0)+y^{\prime}(0)=0, \quad y(2 \pi)=0\) Text Transcription: y“ + lambda y = 0, y(0) + y ‘(0) = 0, y(2 pi) = 0

Write a program, develop in a Fourier-Legendre series, and graph the first five partial sums on common axes, together with the given function. Comment on accuracy. \(e^{2 x} \quad(-1 \leqq x \leqq 1)\) Text Transcription: e^{2x} (-1 leqq x leqq 1)

Write a program, develop in a Fourier-Legendre series, and graph the first five partial sums on common axes, together with the given function. Comment on accuracy. \(\sin \left(\pi x^{2}\right) \quad(-1 \leqq x \leqq 1)\) Text Transcription: sin (pi x^{2}) (-1 leqq x leqq 1)

Write a program, develop in a Fourier-Legendre series, and graph the first five partial sums on common axes, together with the given function. Comment on accuracy. \(1 /(1+|x|) \quad(-1 \leqq x \leqq 1)\) Text Transcription: 1/(1 + |x|) (-1 leqq x leqq 1)

Write a program, develop in a Fourier-Legendre series, and graph the first five partial sums on common axes, together with the given function. Comment on accuracy. \(|\cos \pi x| \quad(-1 \leqq x \leqq 1)\) Text Transcription: |cos pi x| (-1 leqq x leqq 1)

Write a program, develop in a Fourier-Legendre series, and graph the first five partial sums on common axes, together with the given function. Comment on accuracy. x if \(0 \leqq x \leqq 1,0\) if \(-1 \leqq x<0\) Text Transcription: 0 leqq x leqq 1, 0 -1 leqq x < 0