Section C.2 Formula Sheets
A condensed, formula-only companion to the quick references above: each sheet lists the working equations for one unit. Definitions and worked context are left to that unit’s quick reference.
Subsection First-Order Equations
Direct integration. \(\ds\frac{dy}{dx} = f(x)\) gives \(\ds y = \int f(x)\,dx + C\text{.}\)
Separable. \(\ds\frac{dy}{dx} = f(x)\,g(y)\) gives \(\ds\int \frac{dy}{g(y)} = \int f(x)\,dx\text{.}\)
Linear (integrating factor). For \(y' + P(x)\,y = Q(x)\text{,}\)
\begin{equation*}
\mu = e^{\int P(x)\,dx}, \qquad y = \frac{1}{\mu}\int \mu\,Q(x)\,dx.
\end{equation*}
Subsection Qualitative & Numerical Methods
Equilibria. For an autonomous equation \(y' = f(y)\text{,}\) solve \(f(c) = 0\text{.}\) An equilibrium \(c\) is stable when \(f'(c) \lt 0\) and unstable when \(f'(c) \gt 0\text{.}\)
Euler’s method. With step size \(h\) and nodes \(t_k = t_0 + k\,h\text{,}\)
\begin{equation*}
y_{k+1} = y_k + h\,f(t_k, y_k).
\end{equation*}
Accuracy. Local truncation error \(\approx \frac12 y'' h^2\text{;}\) over \(N = (t_N-t_0)/h\) steps the global error \(\left|y_N - y(t_N)\right|\) is proportional to \(h\text{.}\) Euler is first order, so halving \(h\) roughly halves the final error.
Improved Euler (Heun). Predict, then correct with the average of the two slopes:
\begin{align*}
\tilde{y}_{k+1} \amp = y_k + h\,f(t_k,y_k),\\
y_{k+1} \amp = y_k + \frac{h}{2}\Big[f(t_k,y_k) + f\big(t_{k+1},\tilde{y}_{k+1}\big)\Big].
\end{align*}
Second order: halving \(h\) quarters the error.
Subsection Constant-Coefficient Linear Equations
Characteristic equation. For \(a_n y^{(n)} + \cdots + a_1 y' + a_0 y = 0\text{,}\) substituting \(y = e^{rx}\) gives
\begin{equation*}
a_n r^n + \cdots + a_1 r + a_0 = 0.
\end{equation*}
Homogeneous solution, by root type:
\begin{align*}
\text{real, distinct } r_i: \amp\quad y_h = C_1 e^{r_1 x} + \cdots + C_n e^{r_n x},\\
\text{repeated } r \text{ (mult. } m): \amp\quad (C_1 + C_2 x + \cdots + C_m x^{m-1})\,e^{rx},\\
\text{complex } \alpha \pm \beta i: \amp\quad e^{\alpha x}\left(C_1 \cos(\beta x) + C_2 \sin(\beta x)\right).
\end{align*}
Nonhomogeneous (undetermined coefficients). \(y = y_h + y_p\text{,}\) with \(y_p\) matched to the forcing term and multiplied by a power of \(x\) when it overlaps \(y_h\text{.}\)
Subsection Laplace Transforms
Definition. \(\ds \lap{f(t)} = \int_0^\infty e^{-st} f(t)\,dt = F(s)\text{.}\)
Common transforms.
\begin{align*}
\lap{1} \amp= \frac{1}{s}, \amp \lap{e^{at}} \amp= \frac{1}{s-a}, \amp \lap{t^n} \amp= \frac{n!}{s^{n+1}},\\
\lap{\sin(bt)} \amp= \frac{b}{s^2+b^2}, \amp \lap{\cos(bt)} \amp= \frac{s}{s^2+b^2}. \amp \amp
\end{align*}
Transforms of derivatives.
\begin{align*}
\lap{f'(t)} \amp= sF(s) - f(0),\\
\lap{f''(t)} \amp= s^2 F(s) - s f(0) - f'(0).
\end{align*}
Unit step and shifting. \(\ds \lap{u_c(t)} = \frac{e^{-cs}}{s}\text{,}\) and \(\lap{f(t-c)\,u_c(t)} = e^{-cs}F(s)\text{.}\)
Subsection First-Order Linear Systems
Matrix form. \(\vec{X}' = A\vec{X}\text{,}\) with eigenvalues from \(\det(A - rI) = 0\text{.}\)
General solution. For real, distinct eigenvalues \(r_1, r_2\) with eigenvectors \(\vec{v}_1, \vec{v}_2\text{,}\)
\begin{equation*}
\vec{X}(t) = C_1 \vec{v}_1 e^{r_1 t} + C_2 \vec{v}_2 e^{r_2 t}.
\end{equation*}
Subsection Nonlinear Systems
Equilibria. For \(x' = f(x,y)\text{,}\) \(y' = g(x,y)\text{,}\) solve \(f(x^*,y^*) = 0\) and \(g(x^*,y^*) = 0\text{.}\) Factor each equation and test every pairing of cases; never divide by an unknown.
Nullclines. \(f(x,y) = 0\) gives the \(x\)-nullclines (vertical motion); \(g(x,y) = 0\) gives the \(y\)-nullclines (horizontal motion). Equilibria are their crossings, and the signs of \(f\) and \(g\) at one test point fix the direction of flow throughout a region.
Jacobian.
\begin{equation*}
J(x,y) = \begin{bmatrix} f_x \amp f_y \\ g_x \amp g_y \end{bmatrix}, \qquad \det\left(J(x^*,y^*) - rI\right) = 0.
\end{equation*}
Classification. Read the eigenvalues of \(J(x^*,y^*)\) with the linear-system table: both real negative \(\Rightarrow\) stable node; both real positive \(\Rightarrow\) unstable node; real of opposite signs \(\Rightarrow\) saddle; complex with negative real part \(\Rightarrow\) spiral sink; complex with positive real part \(\Rightarrow\) spiral source.
Validity. The classification transfers to the nonlinear system whenever every eigenvalue has nonzero real part. Purely imaginary eigenvalues (a linearized center) or a zero eigenvalue leave the verdict undetermined.
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