Integrating Factor Method
$$\text{1st order linear ODE:}~~\frac{d }{dx}+p(x)y=f(x)$$
$$\mu=e^{\int_{}^{}p(x)dx}$$
$$y\mu=\int_{}^{}\mu f(x)dx +c$$
Wronskian
$$W=\left|\begin{matrix}f_1&f_2&\cdots &f_n\\f_1^{'}&f_2^{'}&\cdots&f_n^{'}\\\vdots\\f_1^{(n-1)}&f_2^{(n-1)}&\cdots&f_n^{(n-1)}\end{matrix}\right|$$
Reduction of Order Method
$$\text{2nd order linear homogeneous equation:}~~\frac{d^2y}{dx^2}+P(x)\frac{dy}{dx}+Q(x)y=0$$
$$y_2=y_1\int_{}^{}\frac{^{-\int_{}^{}P(x)dx}}{y_1^2}dx$$
Characteristic Equation + General Solutions
$$\text{2nd order linear homogeneous equation with constant coefficients:}~~ay^{''}+by^{'}+cy=0$$
$$\text{Characteristic equation:}~~am^2+bm+c=0$$
$$\text{Case 1}~(m_1,m_2~\text{are real and unequal}):~~y=c_1e^{m_1x}+c_2e^{m_2x}$$
$$\text{Case 2}~(m_1=m_2=m):~~y=c_1e^{mx}+c_2xe^{mx}$$
$$\text{Case 3}~(m=\alpha\pm i\beta):~~y=e^{\alpha x}[c_1\cos(\beta x)+c_2\sin(\beta x)]$$
Laplace Transform
$$\mathscr{L}\{f(t)\}=\int_{0}^{\infty}e^{-st}f(t)dt$$
$$\mathscr{L}^{-1}\{F(s)\}=f(t)$$
Laplace Transform of Basic Functions
$$\mathscr{L}\{1\}=\frac{1}{s}$$
$$\mathscr{L}\{t^n\}=\frac{n!}{s^{n+1}}$$
$$\mathscr{L}\{e^{at}\}=\frac{1}{s-a}$$
$$\mathscr{L}\{\sin kt\}=\frac{k}{s^2+k^2}$$
$$\mathscr{L}\{\cos kt\}=\frac{s}{s^2+k^2}$$
Inverse Laplace Transform of Basic Functions
$$\mathscr{L}^{-1}\left\{\frac{1}{s^n}\right\}=\frac{t^{n-1}}{(n-1)!}$$
$$\mathscr{L}^{-1}\left\{\frac{1}{s-a}\right\}=e^{at}$$
$$\mathscr{L}^{-1}\left\{\frac{1}{s^2+k^2}\right\}=\frac{1}{k}\sin kt$$
$$\mathscr{L}^{-1}\left\{\frac{s}{s^2+k^2}\right\}=\cos kt$$
Laplace Transform Translation Theorems
$$\text{First:}~~\mathscr{L}\{e^{at}f(t)\}=F(s-a)=[\mathscr{L}\{f(t)\}]_{s\to(s-a)}$$
$$\text{Inverse:}~~\mathscr{L}^{-1}\{F(s-a)\}=\mathscr{L}^{-1}\{[F(s)]_{s\to(s-a)}\}=e^{at}\mathscr{L}^{-1}\{F(s)\}$$
$$\text{Second:}~~\mathscr{L}\{f(t)\mathscr{U}(t-a)\}=e^{-as}\mathscr{L}\{f(t+a)\}$$
$$\text{Inverse:}~~\mathscr{L}^{-1}\{e^{-as}F(s)\}=f(t-a)\mathscr{U}(t-a)=[\mathscr{L}^{-1}\{F(s)\}]_{t\to(t-a)}\mathscr{U}(t-a)$$
Laplace Transform Differentiation Theorem
$$\mathscr{L}\{t^nf(t)\}=(-1)^n\frac{d^n}{ds^n}\mathscr{L}\{f(t)\},~~n\in\mathbb{N}$$
Laplace Transform of a Derivative
$$\mathscr{L}\{f^n(t)\}=s^nF(s)-s^{n-1}f(0)-s^{n-2}f^{'}(0)-\cdots-f^{(n-1)}(0)$$
Convolution Theorem
$$f*g=\int_{0}^{t}f(\tau)g(t-\tau)d\tau$$
$$\mathscr{L}\{f*g\}=\mathscr{L}\{f(t)\}\mathscr{L}\{g(t)\}=F(s)G(s)$$
Cauchy-Euler Equation + General Solutions
$$a_nx^n\frac{d^ny}{dx^n}+a_{n-1}x^{n-1}\frac{d^{n-1}y}{dx^{n-1}}+\cdots+a_1x\frac{dy}{dx}+a_0y=g(x)$$
$$\text{General, 2nd order, homogeneous Cauchy-Euler equation:}~~ax^2y^{''}+bxy^{'}+cy=0$$
$$\text{Characteristic equation:}~~am^2+(b-a)m+c=0$$
$$\text{Case 1}~(m_1,m_2~\text{are real and unequal}):~~y=c_1x^{m_1}+c_2x^{m_2}$$
$$\text{Case 2}~(m_1=m_2=m):~~y=c_1x^m+c_2x^m\ln{x}$$
$$\text{Case 3}~(m=\alpha\pm i\beta):~~y=x^\alpha[c_1\cos(\beta\ln{x})+c_2\sin(\beta\ln{x})]$$