Question 1 · local series and improper analysis
Use one consistent Taylor expansion throughout. For every limit, state whether it is an endpoint limit or an internal limit and identify the expansion or comparison that justifies it. When the quotient is extended at the apparent singular point, define its value there before discussing differentiability. For the improper integral, write it as the sum of separate integrals at every endpoint or internal point where convergence is not automatic; a single unsupported statement that the integral converges or diverges is not sufficient.
Structure the answer by logical dependencies rather than by isolated formulae. Record intermediate conclusions needed by later parts and check them against the original assumptions.
(a) State Taylor's theorem through order four about x=0, including a valid local remainder. (2 marks)
(b) Expand \(e^x\) about x=0 through fourth order and state the interval or neighbourhood on which the expansion is justified. (4 marks)
(c) On (-∞,∞), define \(g(x)=(e^x-1)/(x)\) away from x=0. Calculate the three limiting regimes -∞,0,∞, state with justification whether the singularity at x=0 is removable, and determine the differentiability of the extension. (7 marks)
(d) State with justification whether the integral of \(g(x)/(1+x^2)\) over its full domain converges as an ordinary improper integral. Split at every endpoint or internal point requiring separate analysis and give comparison functions. (7 marks)
**[Total: 20 marks]**
Worked solution and marking guidance
**(a)** If \(f\) has the required derivatives near \(a\), then
\[
f(x)=\sum_{k=0}^{4}\frac{f^{(k)}(a)}{k!}(x-a)^k+R_4(x),
\]
where, for example under a \(C^5\) hypothesis,
\[
R_4(x)=\frac1{4!}\int_a^x(x-t)^4f^{(5)}(t)\,dt
\]
(or the equivalent Lagrange/Peano local remainder).
**(b)** Put \(h=x-0\). The fourth-order expansion is
\[
e^x=1+x+\frac{x^2}{2}+\frac{x^3}{6}+\frac{x^4}{24}+O(x^5).
\]
The stated neighbourhood/radius must be checked before substituting a
new point.
**(c)** In the three regimes listed in the question, the limits are
\[
0,\ 1,\ +\infty.
\]
The quotient has a removable singularity and is entire after setting \(g(0)=1\).
**(d)** At \(-\infty\) the integrand is \(O(|x|^{-3})\); the point 0 is removable. At \(+\infty\) it behaves like \(e^x/x^3\), so the ordinary improper integral diverges.
Question 2 · conservation, leakage and Fourier data
Before solving, translate the physical description into a volume balance. The liquid occupies an open vessel of constant horizontal area \(S\), its depth is \(h(t)\), and hence its instantaneous volume is \(S h(t)\). Inflow is absent. Outward flow is the sum of a base contribution and a contribution from the wetted porous side wall; both formulae represent volume per time and cease when \(h=0\). Explain the resulting minus sign and preserve the initial condition until the integration constant is known. Include units for the coefficients and, in the requested limiting check, vary only the porous-loss parameter. For the periodic-function part, write down your Fourier convention first, state the convergence value at a discontinuity, and derive the requested scalar series from the resulting coefficient formula rather than quoting it.
Structure the answer by logical dependencies rather than by isolated formulae. Record intermediate conclusions needed by later parts and check them against the original assumptions.
(a) A vertical tank has constant cross-sectional area S=6, initial height \(h(0)=25\), base outflow \(5\sqrt h\), and porous-side outflow \(h\). Derive the conservation equation, verify dimensions, separate it using \(u=\sqrt h\), and express the emptying time in elementary terms. (8 marks)
(b) Introduce a parameter λ multiplying the porous loss. Derive the limit as λ→0 and compare the emptying time with the base-only model, including a physical explanation. (4 marks)
(c) A 2π-periodic signal equals 5 on 0<x<π and 0 on −π<x<0. Compute its Fourier coefficients, state convergence at jumps, and evaluate at a suitable point to derive an alternating odd-reciprocal series identity. (8 marks)
**[Total: 20 marks]**
Worked solution and marking guidance
**(a)** Conservation of volume gives
\[
6\dot h=-5\sqrt h-h.
\]
The coefficient multiplying \(h\) has dimensions area/time. With
\(u=\sqrt h\),
\[
dt=-\frac{12\,du}{5+u},
\]
so the actual two-loss emptying time is
\[
T=12
\log\left(1+\frac{5}{5}\right).
\]
**(b)** After multiplying the porous loss by \(\lambda\),
\[
6\dot h=-5\sqrt h-\lambda h,\qquad
T_\lambda=\frac{12}{\lambda}
\log\left(1+\frac{\lambda\sqrt{25}}{5}\right).
\]
As \(\lambda\to0\),
\[
T_\lambda\longrightarrow T_0
=\frac{12\sqrt{25}}{5}
=12.
\]
Since \(\log(1+x)<x\) for \(x>0\), \(T_\lambda<T_0\): the added
porous loss makes the tank empty sooner.
**(c)** For the signal of height \(A=5\),
\[
a_0=5,\qquad a_n=0,\qquad
b_n=\frac{5}{\pi n}\bigl(1-(-1)^n\bigr).
\]
At a jump the series equals the midpoint \(A/2\). At \(x=\pi/2\),
\[
\frac\pi4=1-\frac13+\frac15-\frac17+\cdots.
\]
Question 3 · linear flow, forcing and exactness
Write the dependent variables as a column vector and show enough work to identify the characteristic polynomial, eigenvalues and a corresponding eigenvector or real fundamental matrix. The phase classification must agree with the large-time behaviour of the solution and with the direction field on a coordinate axis. For the forced system, separate the exponential and constant forcing, test the proposed ansatz for resonance, and then add it to the homogeneous solution. In the differential-form part, identify \(M\) and \(N\), compare the cross-partials on the stated domain, derive rather than guess any integrating factor, and differentiate the recovered potential as a final check.
Structure the answer by logical dependencies rather than by isolated formulae. Record intermediate conclusions needed by later parts and check them against the original assumptions.
(a) For \(X'=AX\), \(A=\begin{pmatrix}4&1\\-2&1\end{pmatrix}\), derive the spectrum and eigenvectors, write a real general solution, classify the origin and describe large-time behaviour. (8 marks)
(b) Add forcing \(F(t)=(e^t,\,5)^T\). Choose a justified particular-solution ansatz, test resonance and state the complete solution. (5 marks)
(c) Consider the differential form \(3x^{3}y\,dx+(x^{4}+15xy^{2})\,dy=0\) on a domain with x≠0. Test exactness; if needed derive a monomial integrating factor and recover a potential function. Verify the result against \(\Phi=x^{3}y+5y^{3}\) only after deriving it. (7 marks)
**[Total: 20 marks]**
Worked solution and marking guidance
**(a)** For \(A=\begin{pmatrix}4&1\\-2&1\end{pmatrix}\), direct calculation gives
\[
\lambda=2,3\), with eigenvectors \(v_2=(1,-2)^T\) and \(v_3=(1,-1)^T\). Hence \(X_h=c_1e^{2t}v_2+c_2e^{3t}v_3\); the origin is an unstable node.
\]
**(b)** The forcing is \((e^t,5)^T\).
The scalar rate is not an eigenvalue in this case, so no resonance
multiplier is required. Solving
\[
(I-A)p=\binom10,\qquad Aq=-\binom0{5}
\]
gives
\[
p=\binom{0}{-1},\qquad q=\binom{5/6}{-10/3}.
\]
Thus \(X=X_h+pe^t+q\). When the exponential rate is zero,
the two constant particular terms may of course be combined.
**(c)** With
\[
M=3x^{3}y,\qquad
N=x^{4}+15xy^{2},
\]
the original cross-partials are unequal. On \(x>0\) or \(x<0\), the
integrating factor \(\mu=x^{-1}\) gives
\[
\widetilde M=3x^{2}y,\qquad
\widetilde N=x^{3}+15y^{2},
\]
whose cross-partials agree. Integrating yields
\[
\Phi(x,y)=x^3y+5y^3=C.
\]
Question 4 · bifurcation, transform and implicit surface
The parameter is allowed to range over all real values for which the vector field is defined. Keep the singular state separate from ordinary equilibria, show the stability of every branch on the parameter intervals between critical values, and label solid, dashed and singular branches on the bifurcation diagram. In the transform problem, state the sign and normalisation convention, transform both differentiation and multiplication by \(x\), and explain why the inverse representation may be written as a real oscillatory integral satisfying the decay requirement. In the implicit problem, evaluate the relevant partial derivative before invoking the implicit-function theorem and then display the full first-order Taylor polynomial at the stated base point.
Mark every critical parameter value on the diagram and state which nearby branches survive on either side of it.
Structure the answer by logical dependencies rather than by isolated formulae. Record intermediate conclusions needed by later parts and check them against the original assumptions.
(a) For \(y'=ry-y^2/(5+y)\), derive every equilibrium and singular state, determine stability for all parameter ranges, draw the bifurcation diagram on a clearly labelled parameter-state diagram, and classify each exchange or collision. (8 marks)
(b) Treat \(u''-3xu=0\), with u decaying at both infinities, by Fourier transform. State a convention, transform both terms, derive and solve the first-order frequency equation, and explain how an admissible real inverse-integral representation is selected. (6 marks)
(c) Let \(F(x,y,z)=e^{xz}+xy-z-1=0\) define z near (0,1,0). Check the implicit-function hypothesis, compute \(z_x,z_y\) at the base point, and write the first-order Taylor approximation. (6 marks)
**[Total: 20 marks]**
Worked solution and marking guidance
**(a)** The vector field factors as
\[
F_r(y)=\frac{y(5r+(r-1)y)}{5+y}.
\]
The singular state is \(y=-5\). The equilibria are \(y=0\)
and, for \(r\ne1\),
\[
y_*=\frac{5r}{1-r}.
\]
Their linearisation derivatives are \(F_r'(0)=r\) and
\(F_r'(y_*)=r(r-1)\). Thus \(0\) is stable for \(r<0\) and unstable
for \(r>0\); \(y_*\) is stable for \(0<r<1\) and unstable for
\(r<0\) or \(r>1\). The branches exchange stability in a
transcritical bifurcation at \(r=0\). At \(r=1\) the second branch
escapes through infinity. The singular line must remain a barrier on
the diagram.
**(b)** With
\[
\widehat u(\xi)=\int_{\mathbb R}u(x)e^{-i\xi x}\,dx,
\]
\[
\mathcal F(u'')=-\xi^2\widehat u,\qquad
\mathcal F(3xu)=3i\widehat u',
\]
so
\[
-\xi^2\widehat u-3i\widehat u'=0,\qquad
\widehat u=C\exp\left(\frac{i\xi^3}{9}\right).
\]
Pairing positive and negative frequencies gives the real oscillatory
integral
\[
u(x)=\frac C\pi\int_0^\infty
\cos\left(\xi x+\frac{\xi^3}{9}\right)\,d\xi,
\]
equivalently a scaled Airy \(Ai\) solution selected by decay.
**(c)** At \((0,1,0)\),
\[
F_x=1,\qquad F_y=0,\qquad F_z=-1\ne0.
\]
Hence the implicit-function theorem applies,
\[
z_x=1,\qquad z_y=0,\qquad
z(x,y)=x+
o\!\left(\sqrt{x^2+(y-1)^2}\right).
\]