Vectors, series & differential equations

The topic on one screen

• Proof by induction has four visible stages: verify the base case, state the assumption for $n=k$, prove the result for $n=k+1$, then conclude for all permitted integers. The step must use the assumption rather than re-proving a special case (CP-1.1).
• Use $\sum r=n(n+1)/2$, $\sum r^2=n(n+1)(2n+1)/6$ and $\sum r^3=[n(n+1)/2]^2$ after expanding a polynomial in $r$ (CP-4.3).
• For differences, rewrite $u_r$ as $f(r+1)-f(r)$ so intermediate terms cancel. Worked example: $\dfrac1{r(r+1)}=\dfrac1r-\dfrac1{r+1}$, hence $\sum_{r=1}^n\dfrac1{r(r+1)}=\dfrac{n}{n+1}$ (CP-4.4).
• A Maclaurin series is $f(x)=\sum_{r=0}^{\infty}\dfrac{f^{(r)}(0)}{r!}x^r$. Derive it when asked; otherwise combine the standard series (CP-4.5).
• The $e^x$, $\sin x$ and $\cos x$ series are valid for every real $x$; $\ln(1+x)$ is valid for $-1<x\le1$; the non-terminating binomial series is valid for $|x|<1$. If the binomial power is a non-negative integer, the series terminates and is an identity for all $x$ (CP-4.6).
• A line in 3-D is $\mathbf r=\mathbf a+\lambda\mathbf d$; its Cartesian form comes from equating $\lambda$. A plane is $\mathbf r\cdot\mathbf n=p$, or a point plus two non-parallel direction vectors (CP-6.1 to CP-6.2).
• The scalar product gives $\mathbf a\cdot\mathbf b=|\mathbf a||\mathbf b|\cos\theta$. It tests perpendicularity and supplies angles between lines, planes and a line and plane. For a line-plane angle $\alpha$, use $\sin\alpha=|\mathbf d\cdot\mathbf n|/(|\mathbf d||\mathbf n|)$ (CP-6.3 to CP-6.4).
• For a line-plane intersection, substitute the line into the plane to find the parameter. For the distance from point $P$ to line $\mathbf r=\mathbf a+\lambda\mathbf d$, solve $[\overrightarrow{OP}-(\mathbf a+\lambda\mathbf d)]\cdot\mathbf d=0$ for the perpendicular foot, then take its distance from $P$. Point-to-plane distance is $|\overrightarrow{OP}\cdot\mathbf n-p|/|\mathbf n|$ (CP-6.5).
• For lines $\mathbf a+\lambda\mathbf d$ and $\mathbf b+\mu\mathbf e$, make their shortest connector $\mathbf c=\mathbf a+\lambda\mathbf d-(\mathbf b+\mu\mathbf e)$ perpendicular to both lines: solve $\mathbf c\cdot\mathbf d=0$ and $\mathbf c\cdot\mathbf e=0$, then calculate $|\mathbf c|$ (CP-6.5).
• For $y'+P(x)y=Q(x)$, the integrating factor is $e^{\int P(x)\,dx}$. Multiplying makes the left side one derivative. Worked example: $y'+2y=e^{-x}$ gives $(e^{2x}y)'=e^x$, so $y=e^{-x}+Ce^{-2x}$ (CP-9.1 to CP-9.2).
• In modelling, define variables and units, translate each rate with the correct sign, solve, then interpret constants and long-term behaviour. For example, a falling particle with linear resistance proportional to speed can give $dv/dt=g-kv$; its terminal speed $g/k$ follows by setting $dv/dt=0$. A plausible equation can still be a poor model if that limit or its assumptions are unrealistic (CP-9.3).
• For $y''+ay'+by=0$, solve the auxiliary equation $m^2+am+b=0$. Distinct real roots give two exponentials, a repeated root gives $(A+Bx)e^{mx}$, and complex roots $p\pm iq$ give $e^{px}(A\cos qx+B\sin qx)$ (CP-9.4 and CP-9.6).
• For a non-homogeneous equation, write $y=\text{CF}+\text{PI}$. Choose the PI trial from the forcing term; if it duplicates part of the CF, multiply the trial by enough powers of $x$ (CP-9.5).
• Simple harmonic motion satisfies $x''=-\omega^2x$ and has solution $x=A\cos\omega t+B\sin\omega t$, with amplitude $\sqrt{A^2+B^2}$ and period $2\pi/\omega$; the acceleration always points towards the equilibrium position. Damping adds a velocity term and makes the auxiliary roots reveal under-, critical or over-damping (CP-9.7 to CP-9.8).
• For coupled first-order equations, eliminate one variable to produce a second-order equation, solve it, then recover the eliminated variable from an original equation. Use every initial condition and substitute back into both equations as a check (CP-9.9).
• Common errors: proving only the $k+1$ case in induction, omitting a series validity range, confusing a plane normal with a line direction, using cosine instead of sine for a line-plane angle, dropping the integration constant, or choosing a PI already contained in the CF.
• Exam technique: separate modelling, solving and interpretation. Define the equation first, keep exact constants until the final line, and use calculator graphs or numerical substitution only as checks on the analytic solution.

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