Back to coursesMATH40004 SIX SOLUTIONS REVISED 2
English review edition prepared on 4 October 2026 from a preserved source copy. It is a later presentation, not the historical study interface. Source checks and difficulty judgements describe the original material author's own process; they do not indicate Imperial College London endorsement.
Source SHA-256: 75a5087fb4c832feddd58436b706e17ec580dbefe5284926ffeaf426548fe439
Source date: 2026-08-06Source page 1
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Mathematical typesetting and diagrams are retained below. Chinese prose is replaced by the English passages that follow, in source reading order.
English passage 1 (13% down the source page)
Recognition signal
Taylor theorem + a quotient whose numerator has a known odd power series. The first decisive step is
to expand arctan x, not to differentiate the quotient repeatedly.
English passage 2 (77% down the source page)
Common errors
Forgetting endpoint convergence of the arctangent series; claiming decimal accuracy without an error theorem;
differentiating g repeatedly instead of reading coefficients.
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English passage 1 (13% down the source page)
Recognition signal
A geometric loss law plus a bulk first-order loss. The key substitution is the geometric length scale
u = V 1/3.
English passage 2 (74% down the source page)
Common errors
Using V ′ = 4πR2R′ and then losing a factor; cubing a negative u after extinction; using the point value rather
than the midpoint value at a Fourier jump.
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English passage 1 (13% down the source page)
Recognition signal
A constant 2 × 2 system: calculate eigenvalues first. For the forcing, split the ansatz into an et part and
a constant part.
English passage 2 (89% down the source page)
Common errors
Calling the spiral unstable because it rotates; using one ansatz for both forcing frequencies; differentiating
xmynM without the product rule.
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English passage 1 (13% down the source page)
Recognition signal
One-dimensional autonomous dynamics: factor the right-hand side, make a sign line for each parameter
range, then place the branches in the bifurcation diagram.
English passage 2 (86% down the source page)
Common errors
Confusing stable/unstable branch line style; forgetting the transform rule xy ↔iby′; solving for zx before
checking Fz̸ = 0.
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English passage 1 (13% down the source page)
Recognition signal
Three independent convergence tests, followed by a binomial-to-antiderivative method and an iterated
chain rule.
English passage 2 (59% down the source page)
Common errors
Using a derivative-heavy route for the binomial series; giving six decimals without controlling the positive tail;
solving L = sin L before proving convergence.
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English passage 1 (13% down the source page)
Recognition signal
Exploit evenness first; for the shifted function use the Fourier shift identity rather than re-integrating.
English passage 2 (62% down the source page)
Common errors
Writing a0 as the constant term instead of a0/2; failing to expand cos(n(x −π/2)); losing the factor 2π in the
convolution theorem.
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English passage 1 (13% down the source page)
Recognition signal
Euler–Cauchy means use z = log x; a repeated eigenvalue with one eigenvector means a Jordan chain.
English passage 2 (82% down the source page)
Common errors
Missing the −Y ′ term in x2y′′; failing to multiply a resonant ansatz by z2; using a second eigenvector when
none exists.
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English passage 1 (13% down the source page)
Recognition signal
Factor the autonomous vector field. At branch intersections, compare stability on both sides before
naming the bifurcation.
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English passage 1 (38% down the source page)
Common errors
Treating every branch meeting as pitchfork; interchanging the coordinates of the saddle points; classifying by
trace alone when the determinant is negative.
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English passage 1 (13% down the source page)
Recognition signal
This question mixes global graph information, nonzero-centre Taylor expansion, a Riemann sum and a
parametric arc-length calculation.
English passage 2 (89% down the source page)
Common errors
Calling the stationary inflection at zero a maximum; using width 1/n instead of π/(4n); forgetting that an infinite
number of turns can have finite length.
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Recognition signal
Volumes use disc/shell formulas; the rod uses mass times centre = moment; Laplace inversion uses partial
fractions.
English passage 2 (59% down the source page)
Common errors
Using the surface-area formula instead of volume; leaving ρ0 in the centre of mass; confusing 1/(s2 + 9) with
the transform of sin 3t without the factor 3.
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English passage 1 (13% down the source page)
Recognition signal
Use oddness for the Fourier coefficients; use periodic boundary terms twice before applying Parseval.
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Common errors
Using the function value at the periodic jump; retaining an a0 term despite zero mean; saying E > 0 at ε = 2
without checking the n = 1 mode.
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English passage 1 (13% down the source page)
Recognition signal
For the system, eigenvectors give the separatrices. For the transform integral, compute the cosine trans-
form first. For Euler–Cauchy, stay in z = log x.
English passage 2 (88% down the source page)
Common errors
Reversing stable and unstable eigenlines; using the cosine transform directly in Plancherel without the factor
two; choosing A = log z instead of integrating twice.
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English passage 1 (13% down the source page)
Recognition signal
At a special point, use the definitions before differentiating the formula for x̸ = 0. Parametric tangents
come from numerator/denominator zero conditions.
English passage 2 (74% down the source page)
Common errors
Using the derivative formula at x = 0; confusing non-continuity of f ′ with non-existence of f ′(0); setting both
numerator and denominator to zero without checking.
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English passage 1 (13% down the source page)
Recognition signal
Split an improper integral at a finite point and analyse each endpoint separately. The centroid uses first
moments divided by area.
English passage 2 (70% down the source page)
Common errors
Testing only infinity; claiming oscillation alone guarantees convergence; using
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English passage 1 (13% down the source page)
Recognition signal
Use Taylor coefficients plus a remainder estimate; use oddness for the sign function; differentiate a
continuous Fourier series only after identifying its piecewise derivative.
English passage 2 (49% down the source page)
Common errors
Giving a decimal without a remainder argument; using a cosine series for an odd function; assigning the one-
sided derivative value at a jump instead of the midpoint.
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English passage 1 (13% down the source page)
Recognition signal
For y′′ = F(y, y′), reduce order by treating u = y′ as a function of y. The delta equation is a standard
transform-domain Green function.
English passage 2 (89% down the source page)
Common errors
Cancelling u without noting the exceptional constant solution; missing the shift phase; saying the origin is the
only fixed point when λ = 0.
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English passage 1 (13% down the source page)
Recognition signal
Flat exponentials defeat every algebraic power; the remaining parts are standard integration by parts,
Riemann sum and integral test.
English passage 2 (35% down the source page)
Common errors
Assuming smoothness merely because the pointwise limit exists; claiming the integral diverges from f →0
without rate; using mesh 1/n.
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English passage 1 (13% down the source page)
Recognition signal
Repeated root + switched forcing: solve before and after the switching time and match both y and y′.
Fourier moments come from differentiating bf.
English passage 2 (46% down the source page)
Common errors
Matching only y; forgetting the forcing vanishes after t = 1; losing the factor i in the third transform derivative.
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English passage 1 (13% down the source page)
Recognition signal
A zero eigenvalue means a line of fixed points. In implicit differentiation, remember that z depends on
x, y again when taking second derivatives.
English passage 2 (43% down the source page)
Common errors
Saying every trajectory tends to the origin; labelling a centre asymptotically stable; treating z as constant in the
second derivative.
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English passage 1 (13% down the source page)
Recognition signal
For a monomial integrating factor, equate powers after differentiating. For multivariable classification,
keep the gradient equations factored.
English passage 2 (82% down the source page)
Common errors
Integrating the unmultiplied equation; expanding the gradient too early and losing roots; classifying a point with
negative determinant as an extremum.
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English passage 1 (13% down the source page)
Recognition signal
Subtract the two logarithm series; divide only after the odd series has been established; prove iteration
convergence before solving the limit equation.
English passage 2 (64% down the source page)
Common errors
Treating endpoint divergence as conditional convergence; using a first omitted term as an error bound for a
positive series without bounding the full tail; solving L = tanh L first.
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English passage 1 (13% down the source page)
Recognition signal
For a wire, use line mass aρ0dθ and moments. For the differentiated Fourier series, first find the contin-
uous even function’s coefficients.
English passage 2 (58% down the source page)
Common errors
Counting the base arc twice in the net moment; expanding sin θ instead of the optimum equation; differentiating
a discontinuous Fourier series without midpoint values.
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English passage 1 (13% down the source page)
Recognition signal
The homogeneous system is a harmonic oscillator. For resonant forcing, multiply the usual sinusoidal
ansatz by t. For an integrating factor depending on y, solve the exactness equation after multiplication.
English passage 2 (89% down the source page)
Common errors
Using a nonresonant ansatz; differentiating xp incorrectly when forming yp; guessing e−y instead of solving the
exactness condition.
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English passage 1 (13% down the source page)
Recognition signal
A singular branch is part of the domain, not a fixed-point branch. In the transform problem, the shift
x −2 becomes a phase translation. For the sphere, use an order-by-order implicit expansion.
English passage 2 (88% down the source page)
Common errors
Calling the singular line a fixed branch; treating the collision as transcritical; losing the −2ω phase; omitting ℓ2