















			<h3 class=blue>Basic ODEs</h3>
			<p>See M1</p>
			<ul>
				<li>Differential equations</li>
				<li>Integrating factor</li>
				<li>First order linear equation</li>
				<li>First order separable equation</li>
				<li>Second order linear, homogeneous, constant coefficients</li>
				<li>Second order linear, constant coefficients</li>
				<li>IVP</li>
				<li>BVP</li>
			</ul>




			<h2 class=yellow>Second order DE</h2>


			<h3 class=yellow>Picard's theorem</h3>



			<h3 class=yellow>Second order linear, homogeneous equations</h3>
			<p>\(y'' + q_1 (x) y' + q_2 (x) y = 0\)</p>
			<ul>
				<li>Obtain a solution \(y_1\)
					<ul><li>Series method</li><li>Cauchy-Euler equation</li><li>Method of Frobenius</li><li>Laplace transfrom method</li><li>Bessel function substitution method</li></ul>
				</li>
				<li>Construct a linearly independent solution \(y_2 = y_1 \int \frac{e^{-\int q_1 (x) dx}}{y_1 (x)^2} dx\]
				\[y(x) = c_1 y_1 (x) + c_2 y_2 (x) \]
			</ul>
			<h4 class=yellow>Second order Cauchy-Euler equation</h4>

			<h3 class=yellow>Second order linear equations</h3>
			<p>\(y'' + q_1 (x) y' + q_2 (x) y = R (x) \)</p>
			<ul>
				<li>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) \]
				<li><b>Variation of parameters</b> implies that we have \(y_p (x) = u(x)y_1 (x) + v(x)y_2 (x)\)
					<ul>
						\[u(x) = - \int \frac{y_2 R (x)}{W(y_1 , y_2)} dx\]
						\[v(x) = \int \frac{y_1 R(x)}{W(y_1 , y_2)} dx\]
					</ul>
				</li>
				\[y(x) = y_c (x) + y_p (x)\]
			</ul>


			<h2 class=red>Series methods</h2>





			<h3 class=red>Airy function</h3>
			<p>Linearly independent solutions to the ODE \(y'' -xy =0\) (Airy equation)</p>
			<h4 class=red>First kind</h4>

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

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

			<h4 class=red>Second kind</h4>
			<p>\(\text{Bi}(x) = \sqrt{3} [ \frac{3^{-\frac{2}{3}}}{\Gamma (\frac{2}{3})} [ 1 + \sum^{\infty}_{n=1} \frac{\prod^{n}_{k=1}(3k-2)}{(3n)!}x^{3n} ] + \frac{3^{-\frac{1}{3}}}{\Gamma (\frac{1}{3})} [ x + \sum^{\infty}_{n=1} \frac{\prod^{n}_{k=1}(3k-1)}{(3n+1)!}x^{3n+1} ] ] \)</p>


			<p>\(\text{Bi}(x) = \frac{1}{\pi} \int^{\infty}_{0} [ \exp ( -\frac{t^3}{3} +xt) + \sin ( \frac{t^3}{3} +xt ) ] dt\)</p>
			<h3 class=red>Regular singular point</h3>

			<h3 class=red>Method of Frobenius</h3>
			<p>Series method variant when singularity at 0</p>
			<ul>
				<li>Assume the ansatz \(y(x) = \sum^{\infty}_{n=0} a_n x^{n+s}\) where \(s \in \mathbb{C}\]
				<li>Substitute the ansatz into the DE and manipulate into one single sum<ul>
				\[ [ \sum^{\infty}_{n=0} a_n x^{n+s} ]^{(k)} = \sum^{\infty}_{n=0} a_n [ \prod^{k-1}_{i=0} (n+s-i) ] x^{n+s-k}\]</ul><li>
				<li>Since the RHS equals zero, equate the coefficient of the new sum to 0</li>
				<li>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</li>
				<li>Solve recursive formula for \(a_n\]
			</ul>
			<h4 class=red>Single solution for \(s\)</h4>
			<p>\(y'' + q_1 (x) y' + q_2 (x) y = 0\)</p>
			<p>\( s_1 \text{ is the only value of} s \text{ and produces solution } y_1(x) \implies \)</p>
			<p>\(y_2(x) = y_1 (x) \ln (x) + x^{s_1} \sum^{\infty}_{n=1}a'_n (s_1)x^n\)</p>
			<h4 class=red>\(s_1 -s_2 \in \mathbb{Z}\)</h4>
			<p>\(y'' + q_1 (x) y' + q_2 (x) y = 0\)</p>
			<p>\( s_1 -s_2 \in \mathbb{Z} \land s_1 \text{ produces solution } y_1(x) \implies \)</p>
			<p>\(y_2(x) = \frac{b_N}{a_0}y_1 (x) \ln (x) + x^{s_2} \sum^{\infty}_{n=0}b'_n (s_2)x^n\)</p>
			<p>\( b_n = (s-s_2) a_n( s) \)</p>



			<h2 class=orange>Bessel functions</h2>




<!--<h3 class=orange>Bessel's equation</h3>
<p>Differential equation that is encountered in solving the Helmholtz equation in polar coordinates</p>
<p>Solution to \(  x^2 \frac{d^2 y}{dx^2} + x \frac{d y}{dx} - (x^2 + \alpha^2) y = 0 \)</p>
<p>Solutions to Bessel's equations found by the method of Frobenius</p>
<p>\(  x^2 \frac{d^2 y}{dx^2} + x \frac{d y}{dx} + (x^2 - \alpha^2) y = 0 \)</p>-->

<h3 class=orange>Bessel function</h3>
<p>Linearly independent solutions to the ODE \(  x^2 y'' + x y' + (x^2 - \alpha^2) y = 0 \) (Bessel equation)</p>
<h4 class=orange>First kind</h4>
<p>\( J_{\alpha} (x) = \sum^{\infty}_{k=0} \frac{(-1)^k  }{k! \Gamma (k + \alpha + 1)}(\frac{x}{2} )^{2k+\alpha} \)</p>
<p>\( J_{n} (x) = \frac{1}{\pi} \int^{\pi}_{0} \cos(nt - x \sin t)dt\)</p>
<p>\(\alpha \notin \mathbb{Z} \implies J_{\alpha} , J_{-\alpha} \text{ are linearly independent solutions to Bessel equation}\)</p>
<p>\( n \in \mathbb{Z} \implies J_{-n}(x) = (-1)^n J_{n} (x) \)</p>
<h4 class=orange>Second kind</h4>
<p>\( Y_{n} (x) = \frac{2}{\pi} [ J_{n}(x)(\gamma + \ln( \frac{x}{2}) ) - \frac{1}{2}\sum^{n-1}_{k=0}\frac{(n-k-1)!(\frac{x}{2})^{2k-n}}{k!} - \frac{1}{2} \sum^{\infty}_{k=0} \frac{(-1)^k [H_k +H_{k+n}](\frac{x}{2})^{2k+n}}{k! (k+n)!} ]\)</p>
<p>\( Y_{\alpha} (x) = \frac{J_{\alpha}(x) \cos ( \alpha x) - J_{-\alpha}(x)}{\sin ( \alpha x)} \)</p>
<p>\(\alpha \notin \mathbb{Z} \implies J_{\alpha} , Y_{\alpha} \text{ are linearly independent solutions to Bessel equation}\)</p>

<h3 class=orange>Modified Bessel function</h3>
<p>Linearly independent solutions to the ODE \(  x^2 y'' + x y' + (x^2 + \alpha^2) y = 0 \) (Modified Bessel equation)</p>
<h4 class=orange>First kind</h4>
<p>\(I_{\alpha}(x) = \sum^{\infty}_{k=0} \frac{1}{k! \Gamma (k+\alpha+1)} (\frac{x}{2})^{2k+\alpha}\)</p>
<p>\(  x^2 \frac{d^2 y}{dx^2} + x \frac{d y}{dx} - (x^2 + \alpha^2) y = 0 \) has general solution \(y(x) = c_1 I_{\alpha}(x) + c_2 I_{-\alpha}(x)\)</p>
<p>\(J_{\alpha} (ix) = i^{\alpha}I_{\alpha}(x)\)</p>
<p>\(\alpha \notin \mathbb{Z} \implies I_{\alpha} , I_{-\alpha} \text{ are linearly independent solutions to modified Bessel equation}\)</p>
<h4 class=orange>Second kind</h4>
<p>\(K_{n}(x)\)</p>
<p>\(K_{\alpha}(x) = \frac{\pi}{2} \frac{I_{-\alpha}(x)- I_{\alpha}(x)}{\sin \alpha \pi}\)</p>
<p>\( n \in \mathbb{Z} \implies I_{n} , K_{n} \text{ are linearly independent solutions to modified Bessel equation}\)</p>

<h3 class=orange>Digamma function</h3>
<p>\(\psi (z) = \frac{d}{dz} \ln \Gamma (z) = \frac{\Gamma' (z)}{\Gamma (z)} \)</p>



