Basic ODEs

See M1

• Differential equations
• Integrating factor
• First order linear equation
• First order separable equation
• Second order linear, homogeneous, constant coefficients
• Second order linear, constant coefficients
• IVP
• BVP

Second order DE

Picard's theorem

Second order linear, homogeneous equations

\(y'' + q_1 (x) y' + q_2 (x) y = 0\)

• Obtain a solution \(y_1\)
  • Series method
  • Cauchy-Euler equation
  • Method of Frobenius
  • Laplace transfrom method
  • Bessel function substitution method
• Construct a linearly independent solution \(y_2 = y_1 \int \frac{e^{-\int q_1 (x) dx}}{y_1 (x)^2} dx\] \[y(x) = c_1 y_1 (x) + c_2 y_2 (x) \]

Second order Cauchy-Euler equation

Second order linear equations

\(y'' + q_1 (x) y' + q_2 (x) y = R (x) \)

• Compute the linearly independent \(y_1 ,y_2\) that solve \(y'' + q_1 (x) y' + q_2 (x) y = 0 \) and let \(y_c (x) = c_1 y_1 (x) +c_2 y_2 (x) \]
• Variation of parameters implies that we have \(y_p (x) = u(x)y_1 (x) + v(x)y_2 (x)\)
  \[u(x) = - \int \frac{y_2 R (x)}{W(y_1 , y_2)} dx\] \[v(x) = \int \frac{y_1 R(x)}{W(y_1 , y_2)} dx\]
\[y(x) = y_c (x) + y_p (x)\]

Series methods

Airy function

Linearly independent solutions to the ODE \(y'' -xy =0\) (Airy equation)

First kind

\(\text{Ai}(x) = \frac{3^{-\frac{2}{3}}}{\Gamma (\frac{2}{3})} [ 1 + \sum^{\infty}_{n=1} \frac{\prod^{n}_{k=1}(3k-2)}{(3n)!}x^{3n} ] - \frac{3^{-\frac{1}{3}}}{\Gamma (\frac{1}{3})} [ x + \sum^{\infty}_{n=1} \frac{\prod^{n}_{k=1}(3k-1)}{(3n+1)!}x^{3n+1} ] \)

\(\text{Ai}(x) = \frac{1}{\pi} \int^{\infty}_{0} \cos ( \frac{t^3}{3} +xt)dt\)

Second kind

\(\text{Bi}(x) = \sqrt{3} [ \frac{3^{-\frac{2}{3}}}{\Gamma (\frac{2}{3})} [ 1 + \sum^{\infty}_{n=1} \frac{\prod^{n}_{k=1}(3k-2)}{(3n)!}x^{3n} ] + \frac{3^{-\frac{1}{3}}}{\Gamma (\frac{1}{3})} [ x + \sum^{\infty}_{n=1} \frac{\prod^{n}_{k=1}(3k-1)}{(3n+1)!}x^{3n+1} ] ] \)

\(\text{Bi}(x) = \frac{1}{\pi} \int^{\infty}_{0} [ \exp ( -\frac{t^3}{3} +xt) + \sin ( \frac{t^3}{3} +xt ) ] dt\)

Regular singular point

Method of Frobenius

Series method variant when singularity at 0

• Assume the ansatz \(y(x) = \sum^{\infty}_{n=0} a_n x^{n+s}\) where \(s \in \mathbb{C}\]
• Substitute the ansatz into the DE and manipulate into one single sum
  \[ [ \sum^{\infty}_{n=0} a_n x^{n+s} ]^{(k)} = \sum^{\infty}_{n=0} a_n [ \prod^{k-1}_{i=0} (n+s-i) ] x^{n+s-k}\]
• Since the RHS equals zero, equate the coefficient of the new sum to 0
• Noting that letting \(a_0 = 0\) leads to the trivial solution, hence the precise value(s) of \(s\) must be determined to make the coefficient of \(x^s\) equal to 0
• Solve recursive formula for \(a_n\]

Single solution for \(s\)

\(y'' + q_1 (x) y' + q_2 (x) y = 0\)

\( s_1 \text{ is the only value of} s \text{ and produces solution } y_1(x) \implies \)

\(y_2(x) = y_1 (x) \ln (x) + x^{s_1} \sum^{\infty}_{n=1}a'_n (s_1)x^n\)

\(s_1 -s_2 \in \mathbb{Z}\)

\(y'' + q_1 (x) y' + q_2 (x) y = 0\)

\( s_1 -s_2 \in \mathbb{Z} \land s_1 \text{ produces solution } y_1(x) \implies \)

\(y_2(x) = \frac{b_N}{a_0}y_1 (x) \ln (x) + x^{s_2} \sum^{\infty}_{n=0}b'_n (s_2)x^n\)

\( b_n = (s-s_2) a_n( s) \)

Bessel functions

Bessel function

Linearly independent solutions to the ODE \( x^2 y'' + x y' + (x^2 - \alpha^2) y = 0 \) (Bessel equation)

First kind

\( J_{\alpha} (x) = \sum^{\infty}_{k=0} \frac{(-1)^k }{k! \Gamma (k + \alpha + 1)}(\frac{x}{2} )^{2k+\alpha} \)

\( J_{n} (x) = \frac{1}{\pi} \int^{\pi}_{0} \cos(nt - x \sin t)dt\)

\(\alpha \notin \mathbb{Z} \implies J_{\alpha} , J_{-\alpha} \text{ are linearly independent solutions to Bessel equation}\)

\( n \in \mathbb{Z} \implies J_{-n}(x) = (-1)^n J_{n} (x) \)

Second kind

\( Y_{n} (x) = \frac{2}{\pi} [ J_{n}(x)(\gamma + \ln( \frac{x}{2}) ) - \frac{1}{2}\sum^{n-1}_{k=0}\frac{(n-k-1)!(\frac{x}{2})^{2k-n}}{k!} - \frac{1}{2} \sum^{\infty}_{k=0} \frac{(-1)^k [H_k +H_{k+n}](\frac{x}{2})^{2k+n}}{k! (k+n)!} ]\)

\( Y_{\alpha} (x) = \frac{J_{\alpha}(x) \cos ( \alpha x) - J_{-\alpha}(x)}{\sin ( \alpha x)} \)

\(\alpha \notin \mathbb{Z} \implies J_{\alpha} , Y_{\alpha} \text{ are linearly independent solutions to Bessel equation}\)

Modified Bessel function

Linearly independent solutions to the ODE \( x^2 y'' + x y' + (x^2 + \alpha^2) y = 0 \) (Modified Bessel equation)

First kind

\(I_{\alpha}(x) = \sum^{\infty}_{k=0} \frac{1}{k! \Gamma (k+\alpha+1)} (\frac{x}{2})^{2k+\alpha}\)

\( x^2 \frac{d^2 y}{dx^2} + x \frac{d y}{dx} - (x^2 + \alpha^2) y = 0 \) has general solution \(y(x) = c_1 I_{\alpha}(x) + c_2 I_{-\alpha}(x)\)

\(J_{\alpha} (ix) = i^{\alpha}I_{\alpha}(x)\)

\(\alpha \notin \mathbb{Z} \implies I_{\alpha} , I_{-\alpha} \text{ are linearly independent solutions to modified Bessel equation}\)

Second kind

\(K_{n}(x)\)

\(K_{\alpha}(x) = \frac{\pi}{2} \frac{I_{-\alpha}(x)- I_{\alpha}(x)}{\sin \alpha \pi}\)

\( n \in \mathbb{Z} \implies I_{n} , K_{n} \text{ are linearly independent solutions to modified Bessel equation}\)

Digamma function

\(\psi (z) = \frac{d}{dz} \ln \Gamma (z) = \frac{\Gamma' (z)}{\Gamma (z)} \)