| Concept / Topic | Description / Formula / Technique |
| :--------------------------------------------- | :---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |
| **Differential Equation** | An equation relating an unknown function and one or more of its derivatives |
| **Ordinary Differential Equation (ODE)** | Depends on one independent variable |
| **Partial Differential Equation (PDE)** | Depends on multiple independent variables |
| **Order** | The highest derivative present in the equation |
| **General linear ODE of $n^{th}$ order** | $f_0(x)y^{(n)} + f_1(x)y^{(n-1)} + f_2(x)y^{(n-2)} + \dots + f_n(x)y = F(x)$ |
| **Explicit Solution** | $y = f(x)$ |
| **Implicit Solution** | $f(x, y) = 0$ |
| **Equation: $\frac{d^n y}{dx^n} = f(x)$** | Perform $n$ integrations to find the solution |
| **General Solution** | Contains $n$ arbitrary constants |
| **Particular Solution** | Contains 0 arbitrary constants (specific solution) |
| **Rectangle of Continuity** | Unique solution exists in interval $I$ |
| **Substitution for $y'' = F(x, y')$** | Substitute $y' = z \text{ and } y'' = \frac{dz}{dx}$to get $\frac{dz}{dx} = F(x, z)$ |
| **Substitution for $y'' = F(y, y')$** | Substitute $y' = p(y)$ |
| **First Order: Separable** | $p(y)y' = q(x) \implies \int p(y)dy = \int q(x)dx$ |
| **First Order: Special Form** | For $y' = f(ax + by + c)$, use substitution $z = ax + by + c$ |
| **First Order: Homogeneous** | For $y' = f(y/x)$, use substitution $u = \frac{y}{x} \implies y = ux$ |
| **First Order: Linear** | $y' + p(x)y = q(x)$Use integrating factor $I = e^{\int p(x) dx}$to solve $\frac{d}{dx}(I \cdot y) = I \cdot q(x)$ |
| **First Order: Bernoulli** | $y' + p(x)y = q(x)y^n$Divide by $y^n$ and substitute $u = y^{1-n}$ |
| **First Order: Exact** | $Mdx + Ndy = 0$ where $M_y = N_x$. Solution is $\phi(x, y) = c$ |
| **Elementary Matrix Determinants** | Interchanging rows: $\vert E_1\vert = -1$. <br>Row scaling: $\vert E_2\vert = k$. <br>Row addition: $\vert E_3\vert = 1
lt;br>$I_3 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \xrightarrow{R_1 \leftrightarrow R_2} E_1 = \begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix} \xrightarrow{R_1 \leftrightarrow R_2} \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} = I_3, \quad $ $\text{so} \quad E_1^{-1} = E_1 = \begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix}$ $I_3 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \xrightarrow{kR_2} E_2 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & k & 0 \\ 0 & 0 & 1 \end{bmatrix} \xrightarrow{\frac{1}{k}R_2} \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} = I_3, \quad$$\text{so} \quad E_2^{-1} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \frac{1}{k} & 0 \\ 0 & 0 & 1 \end{bmatrix}$<br>$I_3 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \xrightarrow{R_1 + kR_2} E_3 = \begin{bmatrix} 1 & k & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \xrightarrow{R_1 - kR_2} \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} = I_3, \quad$ $\text{so} \quad E_3^{-1} = \begin{bmatrix} 1 & -k & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$ |
| **Circle of Proof (Invertibility)** | For $n \times n$ matrix $A$, the following are equivalent: <br>$A$ is invertible; <br>$Ax=b$ has unique solution;<br>$Ax=0$ only has trivial solution;<br>$RREF(A) = I_n$ ([[Reduced row-echelon form]]);$A$ is product of elementary matrices;<br>$rank(A)=n$; $nullity(A)=0lt;br>Cols, rows of A are lin. indep.<br>Cols, rows of A span $R^nlt;br>Cols, rows of A form a basis for $R^n$ |
| **Determinant $2 \times 2$** | $\det A = \begin{vmatrix} a_{11} & a_{12} \vert\vert a_{21} & a_{22} \end{vmatrix} = a_{11}a_{22} - a_{12}a_{21}$ |
| **Determinant $3 \times 3$** | $ A = a_{11} \begin{vmatrix} a_{22} & a_{23} \\ a_{32} & a_{33} \end{vmatrix} - a_{12} \begin{vmatrix} a_{21} & a_{23} \\ a_{31} & a_{33} \end{vmatrix} + a_{13} \begin{vmatrix} a_{21} & a_{22} \\ a_{31} & a_{32} \end{vmatrix}$ |
| **Triangular Matrix Det** | $\det A = a_{11} \cdot a_{22} \dots a_{nn}$ |
| **Properties of $\vert A \vert$** | Interchanging rows/cols: $\vert B \vert = -\vert A \vertlt;br>Multiplying row by $k$: $\vert B \vert = k\vert A \vertlt;br>$\vert kA \vert = k^n\vert A \vertlt;br>$\vert AB \vert = \vert A \vert \cdot \vert B \vert$. $\vert A^{-1} \vert = 1/\vert A \vertlt;br>$\vert A \vert = \vert A^T \vert$ |
| **A, B, D are the same but $D_i = A_i + B_i$** | $\vert D\vert = \vert A\vert + \vert B\vert$ |
| **Vector Space Axioms** | t |
| **Basis** | A set that is both Linearly Independent AND spans the vector space $V$ |
| **Null Space of $A$** | The solution set of the homogeneous system $Ax = 0$ |
| **Rank-Nullity Theorem** | $rank(A) + nullity(A) = n$ |
| **Transition Matrix** | $[x]_C = P_{C \leftarrow B} [x]_B$ |
| **Standard Inner Product** | $\langle \mathbf{x, y} \rangle = x_1y_1 + x_2y_2 + \dots + x_ny_n$ |
| **Norm (Magnitude)** | $\Vert \mathbf{x} \Vert = \sqrt{\langle \mathbf{x, x} \rangle} = \sqrt{x_1^2 + x_2^2 + \dots + x_n^2}$ |
| **Cauchy-Schwarz Inequality** | $\vert \langle \mathbf{u, v} \rangle \vert \leq \Vert \mathbf{u} \Vert \cdot \Vert \mathbf{v} \Vert$ |
| **Orthogonality** | Vectors are orthogonal if $\langle u, v \rangle = 0$ |
| **Gram-Schmidt Process** | $v_1 = x_1$; $v_2 = x_2 - \frac{\langle x_2, v_1 \rangle}{\Vert v_1 \Vert^2} v_1$; $v_i = x_i - \sum_{k=1}^{i-1} \frac{\langle x_i, v_k \rangle}{\Vert v_k \Vert^2} v_k$ |
| **Linear Transformation $T$** | $T(c_1u + c_2v) = c_1T(u) + c_2T(v)$ |
| **Kernel and Range** | $Ker(T) = \{v : T(v)=0\}$. $Rng(T) = \{w : w=T(v)\}$. $rank(T) = \dim(Rng(T))$, $nullity(T) = \dim(Ker(T))$ |
| **Eigenvalue / Char. Poly** | Eigenvalues solve $\det(A - \lambda I) = 0$. Poly: $p(\lambda) = \det(A - \lambda I)$ |
| **Properties of Eigenvalues** | For $A^n$, eigenvalue is $\lambda^n$. For $A^{-1}$, eigenvalue is $1/\lambda$ |
| **Similar Matrices** | $A \sim B$ if $AP = PB$. Similar matrices have same det, rank, char. poly, and eigenvalues |
| **Diagonalization** | $A$ is diagonalizable iff it has $n$ linearly independent eigenvectors. $D = P^{-1}AP$ where $P = [v_1 \vert v_2 \vert \dots \vert v_n]$ |
| **Wronskian $W[f_1, \dots, f_k]$** | Determinant of the function and its first $k-1$ derivatives. If $W \neq 0$ at some point, functions are Linearly Independent |
| **Higher Order: Real Roots** | Distinct $r_1, r_2 \implies e^{r_1t}, e^{r_2t}$. Repeating roots $\implies e^{rt}, te^{rt}, t^2e^{rt}$ |
| **Higher Order: Complex Roots** | Roots $\lambda \pm i\mu \implies y_1 = e^{\lambda t} \cos(\mu t), y_2 = e^{\lambda t} \sin(\mu t)$ |
| **Undetermined Coefficients** | For $F(t) = P_m(t)e^{at}$, guess $y_p = t^s Q_m(t)e^{at}$ where $s$ is multiplicity of $a$ in characteristic roots |
| **Variation of Parameters** | $y_p = c_1(t)y_1 + c_2(t)y_2$ where $c_1'y_1 + c_2'y_2 = 0$ and $c_1'y_1' + c_2'y_2' = g(t)$ |
| **Abel's Theorem** | $W(y_1, y_2)(t) = c e^{-\int p(t) dt}$ |
| **Systems: $X' = AX$** | General solution for distinct eigenvalues: $\mathbf{x} = c_1 \xi^{(1)} e^{r_1t} + \dots + c_n \xi^{(n)} e^{r_nt}$ |
| **Systems: Repeated Roots** | If algebraic multiplicity gt;$ geometric multiplicity, second solution is $\mathbf{x}^{(2)} = \xi te^{\rho t} + \eta e^{\rho t}$ where $(A - \rho I)\eta = \xi$ |
| **Non-homogeneous Systems** | If $A$ is diagonalizable: $\mathbf{X} = P\mathbf{Y}$ where $\mathbf{Y}' = D\mathbf{Y} + P^{-1}\mathbf{g}(t)$ |