Power Series Solutions of ODEs – Programme 8

42 question-and-answer flashcards covering higher derivatives, Leibnitz theorem, power-series and Frobenius methods, Bessel functions, Legendre polynomials, Sturm-Liouville theory, orthogonality and related formulas from Programme 8 lecture notes.

What is Leibnitz’s theorem for the nth derivative of a product y = uv?

y(n) = Σ_{r=0}^n [ nCr · u^(n−r) · v^(r) ]

Using Leibnitz’s theorem, what is the 4th derivative of y = uv?

y(4) = u⁽⁴⁾v + 4u⁽³⁾v′ + 6u″v″ + 4u′v⁽³⁾ + uv⁽⁴⁾

Give the nth derivative of y = e^{ax}.

y(n) = aⁿ e^{ax}

Give the nth derivative of y = x^a (a positive integer, n≤a).

y(n) = a! / (a−n)! · x^{a−n}

State the nth derivative of y = sin(ax).

y(n) = aⁿ sin(ax + nπ/2)

State the nth derivative of y = cos(ax).

y(n) = aⁿ cos(ax + nπ/2)

Give the compact expression for the nth derivative of y = sinh(ax).

y(n) = aⁿ/2 · { [1 + (−1)ⁿ] sinh(ax) + [1 − (−1)ⁿ] cosh(ax) }

What form of trial solution is assumed in Frobenius’ method?

y = x^{c} (a₀ + a₁x + a₂x² + …) with a₀ ≠ 0

In Frobenius’ method, what name is given to the equation arising from the lowest power of x?

The indicial equation

Frobenius Case 1: When do two independent series solutions occur directly?

When the roots c₁ and c₂ of the indicial equation differ by a non-integer.

Frobenius Case 2: How is the complete solution obtained if c₂ = c₁ + n and a coefficient becomes indeterminate at c = c₁?

Set c = c₁; the resulting two series (with arbitrary constants) give the general solution.

Frobenius Case 3: What is done when c₂ = c₁ + n and a coefficient becomes infinite at c = c₁?

Replace a₀ by k(c−c₁); differentiate wrt c and set c = c₁ to get the second independent solution containing ln x.

Frobenius Case 4: What is the solution form when the indicial roots are equal (c₁ = c₂)?

y = x^{c}[ (1 + k ln x)(a₀ + a₁x + …) + x^{0}(b₁x + b₂x² + …) ]

Write Bessel’s differential equation of order v.

x²y'' + xy' + (x² − v²) y = 0

What are the two linearly independent Bessel functions of the first kind?

Jv(x) and J{−v}(x)

Give the series definition of the Bessel function J_v(x).

Jv(x) = Σ{k=0}^∞ [ (−1)^k / (k! Γ(k+v+1)) ] · (x/2)^{2k+v}

For integer n, how are J{−n}(x) and Jn(x) related?

J{−n}(x) = (−1)^n Jn(x)

Write Legendre’s differential equation.

(1−x²) y'' − 2x y' + n(n+1) y = 0

State Rodrigue’s formula for Legendre polynomials.

P_n(x) = (1 / 2^n n!) · d^n/dx^n (x² − 1)^n

What is the generating function of Legendre polynomials?

1/√(1−2xt+t²) = Σ{n=0}^∞ Pn(x) t^n, |t|<1

Give P₂(x) and P₃(x).

P₂(x) = ½(3x²−1); P₃(x) = ½(5x³−3x)

State the orthogonality condition for Legendre polynomials.

∫{−1}^{1} Pm(x) P_n(x) dx = 0 for m ≠ n

Define a Sturm-Liouville problem in standard form.

(p y')' + (q + λ r)y = 0 on [a,b] with r>0 and boundary conditions a₁y + a₂y' =0 at x=a, β₁y + β₂y' =0 at x=b

What property do eigenfunctions of a Sturm-Liouville system satisfy?

They are orthogonal with respect to the weight function r(x): ∫ ym yn r dx = 0 for m≠n.

For y'' + λy = 0, y(0)=y(5)=0, give the eigenvalues.

λ_n = n²π² / 25 , n = 1,2,3,…

Write the corresponding eigenfunctions for the previous problem.

yn(x) = An sin(nπx/5)

Express x² as a finite sum of Legendre polynomials.

x² = ½ P₀(x) + ½ P₂(x)

Express 1 + x + x³ in Legendre polynomials.

1 + x + x³ = P₀(x) + ⅓ P₁(x) + 2/3 P₃(x)

What is the nth derivative of y = ln x?

y(n) = (−1)^{n−1} (n−1)! / x^n

Provide the recurrence relation obtained for (1+x²)y'' − 3xy' −5y = 0 about x=0.

y{n+2} = (n(n−1)+3n+5) yn / (n+2)(n+1)

For y = sinh(ax), what is y'(n) when n is odd?

y(n) = aⁿ cosh(ax) when n is odd (since sinh term vanishes)

What is the coefficient pattern in Leibnitz derivatives identified in the notes?

The numerical coefficients are the binomial coefficients.

Write the ratio test criterion for convergence of a power series Σ a_k x^k.

Limit |a{k+1}/ak| |x| < 1 for convergence.

State Jo(x) and J₁(x) first three non-zero terms.

Jo(x) ≈ 1 − x²/4 + x⁴/64 … ; J₁(x) ≈ x/2 − x³/16 + x⁵/384 …

What is the weight function associated with Legendre polynomials in Sturm-Liouville form?

w(x) = 1 on [−1, 1]

How is a polynomial of degree n expressed using Legendre polynomials?

f(x) = Σ{k=0}^{n} ak P_k(x) with finite sum

Give the recurrence relation for Bessel coefficients (from Frobenius).

a{r} = −a{r−2} / [ (v+r−1)(v+r) − v² ] = a_{r−2}/[v² − (v+r)²]

What condition on P(x), Q(x) allows Frobenius at x=0?

xP(x) and x²Q(x) finite ⇒ regular singular point

When does the Frobenius series reduce to elementary functions?

When one of the series terminates after finite terms (e.g., integer order in Legendre or Bessel).

What is the gamma function identity used to re-express Bessel coefficients?

Γ(z+1) = z Γ(z)