See M1
\( y'+q_1 (x) y= q_2 (x) y^{n} \)
\( q_1 (x)y' +q_2 (x)y + q_3 (x)y^{2}= q_4 (x) \)
\(y_4 = \frac{y_1 (y_3 - y_2) + a y_2 (y_1 -y_3)}{y_3 -y_2 + a(y_1 - y_3)}\)
\( P(x,y) dx + Q(x,y)dy = 0 \)
\( y' = \frac{P(x,y)}{Q(x,y)} \)
\( P(x,y) dx + Q(x,y)dy = 0 \land \frac{\partial Q}{\partial x} = \frac{\partial P}{\partial y} \implies \exists F(x,y) ( Q = \frac{\partial F}{\partial y} \land P = \frac{\partial F}{ \partial y} ) \land y(x) : F(x,y)=C \text{ satisfies the DE}\)
\(P,Q \text{ are homogenous of same degree } \implies y=xv(x) \text{ makes ODE separable} \)
Notice how this substitution abuses homogeneity to cancell out the dependent variable \(x\)
Method of forming an ansatz to reduce differential equations into algebraic equations.
| \(g(x)\) | \(y(x)\) |
|---|---|
| \(ke^{ax}\) | \(Ce^{ax}\) |
| \(k \sin (ax)\) | \(K \cos (ax) + M \sin (ax)\) |
| \(k x^n\) | \( \sum^{n}_{i=0} K_i x^i \) |
Unary operator (function) defined in terms of functions and differentials
\( \sum^{n}_{i=1} a_i y^{(i)} = 0 \land \text{dim}(F)=n \land \forall f \in F ( \sum^{n}_{i=1} a_i f^{(i)}=0 ) \implies y(x)= \sum^{n}_{i=1} c_i f_{i}(x) \text{ is the general solution} \)
Function \(W\) which is the Determinant of the matrix where each column is a vector with a solution to a linear DE and its \(n-1\) derivatives.
\(W(\textbf{y}) = \begin{vmatrix} y_1 & y_2 & \cdots & y_n \\ y'_1 & y'_2 & \cdots & y'_n \\ \vdots & \vdots & \ddots & \vdots \\ y^{(n-1)}_{1} & y^{(n-1)}_{2} & \cdots & y^{(n-1)}_{n} \end{vmatrix}\)
\(y_1,y_2 \text{ satisfy } y'' + q_1 (x) y' + q_2 (x) y = 0 \implies W(y_1 , y_2) = K_{12}e^{-\int q_1 (x) dx}\)
\(y_1 \text{ satisfies } y'' + q_1 (x) y' + q_2 (x) y = 0 \land \neg ( \forall x [y_1(x)=0] ) \implies y_2 = y_1 \int \frac{e^{-\int q_1 (x) dx}}{y_1 (x)^2} dx \text{ satisfies } y'' + q_1 (x) y' + q_2 (x) y = 0 \land y_1 , y_2 \text{ are linearly independent} \)
Assuming the ansatz \(y(x) = u(x)y_1(x) + v(x) y_2 (x)\) and \(u' y_1 + v' y_2 = 0\)
\(y'' + q_1 (x) y' + q_2 (x) y = 0\)
\(y'' + q_1 (x) y' + q_2 (x) y = R (x) \)
For differential equations such that \(y^{(n)} + \sum q_k (x) y^{(k)} = 0 \land q_k \text{ are analytic on }I \land 0 \in I\), solutions are always analytic and hence has a Taylor series representation and hence a power series representation. This is due to considering the DE on a complex neighborhood and by noting that holomorphic functions are analytic, see Complex Analysis
\(x_0 \text{ is an ordinary point of } \sum q_n (x) y^{n} (x) = 0 \iff \forall k \in \{0,1,..,n\}a_k \text{ is analytic at }x_0)\)
\( \prod^{n}_{k=1} 2k = 2^n n!\)
\( \prod^{n}_{k=1} (2k-1) = \frac{(2n-1)!}{2^(n-1) (n-1)!}\)
\(x_0 \text{ is a singular point of } \sum q_n (x) y^{n} (x) = 0 \iff \neg [ x_0 \text{ is a singular point of } \sum q_n (x) y^{n} (x) = 0]\)
All coefficient function analytic at a point means all DEs solutions taylor series equal function itself
Linearly independent solutions to the ODE \(y'' -xy =0\) (Airy equation)
\(\text{Ai}(x) = \frac{3^{-\frac{2}{3}}}{\Gamma (\frac{2}{3})} [ 1 + \sum^{\infty}_{n=1} \frac{\prod^{n}_{k=1}(3k-2)}{(3n)!}x^{3n} ] - \frac{3^{-\frac{1}{3}}}{\Gamma (\frac{1}{3})} [ x + \sum^{\infty}_{n=1} \frac{\prod^{n}_{k=1}(3k-1)}{(3n+1)!}x^{3n+1} ] \)
\(\text{Ai}(x) = \frac{1}{\pi} \int^{\infty}_{0} \cos ( \frac{t^3}{3} +xt)dt\)
\(\text{Bi}(x) = \sqrt{3} [ \frac{3^{-\frac{2}{3}}}{\Gamma (\frac{2}{3})} [ 1 + \sum^{\infty}_{n=1} \frac{\prod^{n}_{k=1}(3k-2)}{(3n)!}x^{3n} ] + \frac{3^{-\frac{1}{3}}}{\Gamma (\frac{1}{3})} [ x + \sum^{\infty}_{n=1} \frac{\prod^{n}_{k=1}(3k-1)}{(3n+1)!}x^{3n+1} ] ] \)
\(\text{Bi}(x) = \frac{1}{\pi} \int^{\infty}_{0} [ \exp ( -\frac{t^3}{3} +xt) + \sin ( \frac{t^3}{3} +xt ) ] dt\)
Series method variant when singularity at 0
\(y'' + q_1 (x) y' + q_2 (x) y = 0\)
\( s_1 \text{ is the only value of} s \text{ and produces solution } y_1(x) \implies \)
\(y_2(x) = y_1 (x) \ln (x) + x^{s_1} \sum^{\infty}_{n=1}a'_n (s_1)x^n\)
\(y'' + q_1 (x) y' + q_2 (x) y = 0\)
\( s_1 -s_2 \in \mathbb{Z} \land s_1 \text{ produces solution } y_1(x) \implies \)
\(y_2(x) = \frac{b_N}{a_0}y_1 (x) \ln (x) + x^{s_2} \sum^{\infty}_{n=0}b'_n (s_2)x^n\)
\( b_n = (s-s_2) a_n( s) \)
Linearly independent solutions to the ODE \( x^2 y'' + x y' + (x^2 - \alpha^2) y = 0 \) (Bessel equation)
\( J_{\alpha} (x) = \sum^{\infty}_{k=0} \frac{(-1)^k }{k! \Gamma (k + \alpha + 1)}(\frac{x}{2} )^{2k+\alpha} \)
\( J_{n} (x) = \frac{1}{\pi} \int^{\pi}_{0} \cos(nt - x \sin t)dt\)
\(\alpha \notin \mathbb{Z} \implies J_{\alpha} , J_{-\alpha} \text{ are linearly independent solutions to Bessel equation}\)
\( n \in \mathbb{Z} \implies J_{-n}(x) = (-1)^n J_{n} (x) \)
\( Y_{n} (x) = \frac{2}{\pi} [ J_{n}(x)(\gamma + \ln( \frac{x}{2}) ) - \frac{1}{2}\sum^{n-1}_{k=0}\frac{(n-k-1)!(\frac{x}{2})^{2k-n}}{k!} - \frac{1}{2} \sum^{\infty}_{k=0} \frac{(-1)^k [H_k +H_{k+n}](\frac{x}{2})^{2k+n}}{k! (k+n)!} ]\)
\( Y_{\alpha} (x) = \frac{J_{\alpha}(x) \cos ( \alpha x) - J_{-\alpha}(x)}{\sin ( \alpha x)} \)
\(\alpha \notin \mathbb{Z} \implies J_{\alpha} , Y_{\alpha} \text{ are linearly independent solutions to Bessel equation}\)
Linearly independent solutions to the ODE \( x^2 y'' + x y' + (x^2 + \alpha^2) y = 0 \) (Modified Bessel equation)
\(I_{\alpha}(x) = \sum^{\infty}_{k=0} \frac{1}{k! \Gamma (k+\alpha+1)} (\frac{x}{2})^{2k+\alpha}\)
\( x^2 \frac{d^2 y}{dx^2} + x \frac{d y}{dx} - (x^2 + \alpha^2) y = 0 \) has general solution \(y(x) = c_1 I_{\alpha}(x) + c_2 I_{-\alpha}(x)\)
\(J_{\alpha} (ix) = i^{\alpha}I_{\alpha}(x)\)
\(\alpha \notin \mathbb{Z} \implies I_{\alpha} , I_{-\alpha} \text{ are linearly independent solutions to modified Bessel equation}\)
\(K_{n}(x)\)
\(K_{\alpha}(x) = \frac{\pi}{2} \frac{I_{-\alpha}(x)- I_{\alpha}(x)}{\sin \alpha \pi}\)
\( n \in \mathbb{Z} \implies I_{n} , K_{n} \text{ are linearly independent solutions to modified Bessel equation}\)
\(\psi (z) = \frac{d}{dz} \ln \Gamma (z) = \frac{\Gamma' (z)}{\Gamma (z)} \)
The characteristic function of real numbers greater than or equal to 0, \(\chi_{\mathbb{R}_{+}}\), it serves useful for the Laplace inversion of function with a factor \(e^{-sa}\)
\(H(t) = \begin{cases} 1 & t \geq 0 \\ 0 & t \lt 0 \end{cases}\)
Integral transform on \((0,\infty)\) with kernel \(e^{-st}\)
\( \displaystyle \mathcal{L} \{ f \} (s) = \int^{\infty}_{0} f(t) e^{-st} dt \)
| \(f(t)\) | \(\mathcal{L}\{f\}(s)\) |
|---|---|
| \(t^n\) | \( \frac{\Gamma (n +1 )}{s^{n+1}}\) |
| \(e^{-at}\) | \( \frac{1}{s+a}\) |
| \(\sin (at)\) | \( \frac{a}{s^2 + a^2}\) |
| \(\cos (at)\) | \( \frac{s}{s^2 + a^2} \) |
| \( J_0 (t)\) | \( \frac{1}{\sqrt{1 +s^2}} \) |
Since a property of Laplace transforms is that two functions with the same transform must be equal almost everywhere, Laplace transforms are unique and hence invertible
\( \mathcal{L}^{-1} \{ \mathcal{L}\{f\} \} = f \)
\( \mathcal{L}^{-1} \{ af+bg \} = a \mathcal{L}^{-1}\{f\} + b \mathcal{L}^{-1}\{g\} \)
Convert the Laplace transform into a series, use linearity of inverse Laplace operator to resolve each term directly
\(\mathcal{L}\{f * g\}(s) = \mathcal{L}\{f\}(s) \mathcal{L}\{g\}(s)\)
Linearly independent solutions to the ODE \(zw''+(b-z)w'-aw=0\) (Kummer's equation)
\(M(a,b,z)= \sum^{\infty}_{n=0} \frac{a^{(n)} z^n }{ b^{(n)} n! }\)
\(U(a,b,z)\)
Applying the ansatz \(f(x,y)=X(x)Y(y) \) to reduce a PDE to ODEs.
This works when the partial derivatives are known to reduce to constants rather than more complex functions.
See Lebesgue Integration and Fourier Analysis
See Lebesgue Integration and Fourier Analysis
See Lebesgue Integration and Fourier Analysis
Principle that functions harmonic on a closure have their maximum on the boundary of the neighborhood
\(\forall \omega \in \Omega \subset \mathbb{R}^n [ \nabla^2 f(\omega) ] = 0 \implies \text{argmax}_{\omega \in \Omega} \{ f(\omega): \omega \in \Omega\} \in \partial \omega\)