• # Paper 3, Section II, F

Define the terms connected and path-connected for a topological space. Prove that the interval $[0,1]$ is connected and that if a topological space is path-connected, then it is connected.

Let $X$ be an open subset of Euclidean space $\mathbb{R}^{n}$. Show that $X$ is connected if and only if $X$ is path-connected.

Let $X$ be a topological space with the property that every point has a neighbourhood homeomorphic to an open set in $\mathbb{R}^{n}$. Assume $X$ is connected; must $X$ be also pathconnected? Briefly justify your answer.

Consider the following subsets of $\mathbb{R}^{2}$ :

$\begin{gathered} A=\{(x, 0): x \in(0,1]\}, \quad B=\{(0, y): y \in[1 / 2,1]\}, \text { and } \\ C_{n}=\{(1 / n, y): y \in[0,1]\} \text { for } n \geqslant 1 \end{gathered}$

Let

$X=A \cup B \cup \bigcup_{n \geqslant 1} C_{n}$

with the subspace topology. Is $X$ path-connected? Is $X$ connected? Justify your answers.

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• # Paper 3, Section II, G

Let $\gamma$ be a curve (not necessarily closed) in $\mathbb{C}$ and let $[\gamma]$ denote the image of $\gamma$. Let $\phi:[\gamma] \rightarrow \mathbb{C}$ be a continuous function and define

$f(z)=\int_{\gamma} \frac{\phi(\lambda)}{\lambda-z} d \lambda$

for $z \in \mathbb{C} \backslash[\gamma]$. Show that $f$ has a power series expansion about every $a \notin[\gamma]$.

Using Cauchy's Integral Formula, show that a holomorphic function has complex derivatives of all orders. [Properties of power series may be assumed without proof.] Let $f$ be a holomorphic function on an open set $U$ that contains the closed disc $\bar{D}(a, r)$. Obtain an integral formula for the derivative of $f$ on the open disc $D(a, r)$ in terms of the values of $f$ on the boundary of the disc.

Show that if holomorphic functions $f_{n}$ on an open set $U$ converge locally uniformly to a holomorphic function $f$ on $U$, then $f_{n}^{\prime}$ converges locally uniformly to $f^{\prime}$.

Let $D_{1}$ and $D_{2}$ be two overlapping closed discs. Let $f$ be a holomorphic function on some open neighbourhood of $D=D_{1} \cap D_{2}$. Show that there exist open neighbourhoods $U_{j}$ of $D_{j}$ and holomorphic functions $f_{j}$ on $U_{j}, j=1,2$, such that $f(z)=f_{1}(z)+f_{2}(z)$ on $U_{1} \cap U_{2}$.

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• # Paper 3, Section I, B

Find the value of $A$ for which the function

$\phi(x, y)=x \cosh y \sin x+A y \sinh y \cos x$

satisfies Laplace's equation. For this value of $A$, find a complex analytic function of which $\phi$ is the real part.

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• # Paper 3, Section II, 15D

(a) The energy density stored in the electric and magnetic fields $\mathbf{E}$ and $\mathbf{B}$ is given by

$w=\frac{\epsilon_{0}}{2} \mathbf{E} \cdot \mathbf{E}+\frac{1}{2 \mu_{0}} \mathbf{B} \cdot \mathbf{B}$

Show that, in regions where no electric current flows,

$\frac{\partial w}{\partial t}+\boldsymbol{\nabla} \cdot \mathbf{S}=0$

for some vector field $\mathbf{S}$ that you should determine.

(b) The coordinates $x^{\prime \mu}=\left(c t^{\prime}, \mathbf{x}^{\prime}\right)$ in an inertial frame $\mathcal{S}^{\prime}$ are related to the coordinates $x^{\mu}=(c t, \mathbf{x})$ in an inertial frame $\mathcal{S}$ by a Lorentz transformation $x^{\prime \mu}=\Lambda_{\nu}^{\mu} x^{\nu}$, where

$\Lambda_{\nu}^{\mu}=\left(\begin{array}{cccc} \gamma & -\gamma v / c & 0 & 0 \\ -\gamma v / c & \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right)$

with $\gamma=\left(1-v^{2} / c^{2}\right)^{-1 / 2}$. Here $v$ is the relative velocity of $\mathcal{S}^{\prime}$ with respect to $\mathcal{S}$ in the x-direction.

In frame $\mathcal{S}^{\prime}$, there is a static electric field $\mathbf{E}^{\prime}\left(\mathbf{x}^{\prime}\right)$ with $\partial \mathbf{E}^{\prime} / \partial t^{\prime}=0$, and no magnetic field. Calculate the electric field $\mathbf{E}$ and magnetic field $\mathbf{B}$ in frame $\mathcal{S}$. Show that the energy density in frame $\mathcal{S}$ is given in terms of the components of $\mathbf{E}^{\prime}$ by

$w=\frac{\epsilon_{0}}{2}\left[E_{x}^{\prime 2}+\left(\frac{c^{2}+v^{2}}{c^{2}-v^{2}}\right)\left(E_{y}^{\prime 2}+E_{z}^{\prime 2}\right)\right]$

Use the fact that $\partial w / \partial t^{\prime}=0$ to show that

$\frac{\partial w}{\partial t}+\nabla \cdot\left(w v \mathbf{e}_{x}\right)=0$

where $\mathbf{e}_{x}$ is the unit vector in the $x$-direction.

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• # Paper 3, Section I, A

A two-dimensional flow $\mathbf{u}=(u, v)$ has a velocity field given by

$u=\frac{x^{2}-y^{2}}{\left(x^{2}+y^{2}\right)^{2}} \quad \text { and } \quad v=\frac{2 x y}{\left(x^{2}+y^{2}\right)^{2}}$

(a) Show explicitly that this flow is incompressible and irrotational away from the origin.

(b) Find the stream function for this flow.

(c) Find the velocity potential for this flow.

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• # Paper 3, Section II, A

A two-dimensional layer of viscous fluid lies between two rigid boundaries at $y=\pm L_{0}$. The boundary at $y=L_{0}$ oscillates in its own plane with velocity $\left(U_{0} \cos \omega t, 0\right)$, while the boundary at $y=-L_{0}$ oscillates in its own plane with velocity $\left(-U_{0} \cos \omega t, 0\right)$. Assume that there is no pressure gradient and that the fluid flows parallel to the boundary with velocity $(u(y, t), 0)$, where $u(y, t)$ can be written as $u(y, t)=\operatorname{Re}\left[U_{0} f(y) \exp (i \omega t)\right]$.

(a) By exploiting the symmetry of the system or otherwise, show that

$f(y)=\frac{\sinh [(1+i) \Delta \hat{y}]}{\sinh [(1+i) \Delta]}, \text { where } \hat{y}=\frac{y}{L_{0}} \text { and } \Delta=\left(\frac{\omega L_{0}^{2}}{2 \nu}\right)^{1 / 2}$

(b) Hence or otherwise, show that

where $\Delta_{\pm}=\Delta(1 \pm \hat{y})$.

(c) Show that, for $\Delta \ll 1$,

$u(y, t) \simeq \frac{U_{0} y}{L_{0}} \cos \omega t$

and briefly interpret this result physically.

\begin{aligned} & \frac{u(y, t)}{U_{0}}=\frac{\cos \omega t\left[\cosh \Delta_{+} \cos \Delta_{-}-\cosh \Delta_{-} \cos \Delta_{+}\right]}{(\cosh 2 \Delta-\cos 2 \Delta)} \\ & +\frac{\sin \omega t\left[\sinh \Delta_{+} \sin \Delta_{-}-\sinh \Delta_{-} \sin \Delta_{+}\right]}{(\cosh 2 \Delta-\cos 2 \Delta)}, \end{aligned}

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• # Paper 3, Section I, E

State the local Gauss-Bonnet theorem for geodesic triangles on a surface. Deduce the Gauss-Bonnet theorem for closed surfaces. [Existence of a geodesic triangulation can be assumed.]

Let $S_{r} \subset \mathbb{R}^{3}$ denote the sphere with radius $r$ centred at the origin. Show that the Gauss curvature of $S_{r}$ is $1 / r^{2}$. An octant is any of the eight regions in $S_{r}$ bounded by arcs of great circles arising from the planes $x=0, y=0, z=0$. Verify directly that the local Gauss-Bonnet theorem holds for an octant. [You may assume that the great circles on $S_{r}$ are geodesics.]

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• # Paper 3, Section II, E

Let $S \subset \mathbb{R}^{3}$ be an embedded smooth surface and $\gamma:[0,1] \rightarrow S$ a parameterised smooth curve on $S$. What is the energy of $\gamma$ ? By applying the Euler-Lagrange equations for stationary curves to the energy function, determine the differential equations for geodesics on $S$ explicitly in terms of a parameterisation of $S$.

If $S$ contains a straight line $\ell$, prove from first principles that each segment $[P, Q] \subset \ell$ (with some parameterisation) is a geodesic on $S$.

Let $H \subset \mathbb{R}^{3}$ be the hyperboloid defined by the equation $x^{2}+y^{2}-z^{2}=1$ and let $P=\left(x_{0}, y_{0}, z_{0}\right) \in H$. By considering appropriate isometries, or otherwise, display explicitly three distinct (as subsets of $H$ ) geodesics $\gamma: \mathbb{R} \rightarrow H$ through $P$ in the case when $z_{0} \neq 0$ and four distinct geodesics through $P$ in the case when $z_{0}=0$. Justify your answer.

Let $\gamma: \mathbb{R} \rightarrow H$ be a geodesic, with coordinates $\gamma(t)=(x(t), y(t), z(t))$. Clairaut's relation asserts $\rho(t) \sin \psi(t)$ is constant, where $\rho(t)=\sqrt{x(t)^{2}+y(t)^{2}}$ and $\psi(t)$ is the angle between $\dot{\gamma}(t)$ and the plane through the point $\gamma(t)$ and the $z$-axis. Deduce from Clairaut's relation that there exist infinitely many geodesics $\gamma(t)$ on $H$ which stay in the half-space $\{z>0\}$ for all $t \in \mathbb{R}$.

[You may assume that if $\gamma(t)$ satisfies the geodesic equations on $H$ then $\gamma$ is defined for all $t \in \mathbb{R}$ and the Euclidean norm $\|\dot{\gamma}(t)\|$ is constant. If you use a version of the geodesic equations for a surface of revolution, then that should be proved.]

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• # Paper 3, Section I, G

Let $G$ be a finite group, and let $H$ be a proper subgroup of $G$ of index $n$.

Show that there is a normal subgroup $K$ of $G$ such that $|G / K|$ divides $n$ ! and $|G / K| \geqslant n$.

Show that if $G$ is non-abelian and simple, then $G$ is isomorphic to a subgroup of $A_{n}$.

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• # Paper 3, Section II, 10G

Let $p$ be a non-zero element of a Principal Ideal Domain $R$. Show that the following are equivalent:

(i) $p$ is prime;

(ii) $p$ is irreducible;

(iii) $(p)$ is a maximal ideal of $R$;

(iv) $R /(p)$ is a field;

(v) $R /(p)$ is an Integral Domain.

Let $R$ be a Principal Ideal Domain, $S$ an Integral Domain and $\phi: R \rightarrow S$ a surjective ring homomorphism. Show that either $\phi$ is an isomorphism or $S$ is a field.

Show that if $R$ is a commutative ring and $R[X]$ is a Principal Ideal Domain, then $R$ is a field.

Let $R$ be an Integral Domain in which every two non-zero elements have a highest common factor. Show that in $R$ every irreducible element is prime.

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• # Paper 3, Section II, 9E

(a) (i) State the rank-nullity theorem.

Let $U$ and $W$ be vector spaces. Write down the definition of their direct sum $U \oplus W$ and the inclusions $i: U \rightarrow U \oplus W, j: W \rightarrow U \oplus W$.

Now let $U$ and $W$ be subspaces of a vector space $V$. Define $l: U \cap W \rightarrow U \oplus W$ by $l(x)=i x-j x .$

Describe the quotient space $(U \oplus W) / \operatorname{Im}(l)$ as a subspace of $V$.

(ii) Let $V=\mathbb{R}^{5}$, and let $U$ be the subspace of $V$ spanned by the vectors

$\left(\begin{array}{c} 1 \\ 2 \\ -1 \\ 1 \\ 1 \end{array}\right),\left(\begin{array}{l} 1 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right),\left(\begin{array}{c} -2 \\ 2 \\ 2 \\ 1 \\ -2 \end{array}\right)$

and $W$ the subspace of $V$ spanned by the vectors

$\left(\begin{array}{c} 3 \\ 2 \\ -3 \\ 1 \\ 3 \end{array}\right),\left(\begin{array}{l} 1 \\ 1 \\ 0 \\ 0 \\ 0 \end{array}\right),\left(\begin{array}{c} 1 \\ -4 \\ -1 \\ -2 \\ 1 \end{array}\right)$

Determine the dimension of $U \cap W$.

(b) Let $A, B$ be complex $n$ by $n$ matrices with $\operatorname{rank}(B)=k$.

Show that $\operatorname{det}(A+t B)$ is a polynomial in $t$ of degree at most $k$.

Show that if $k=n$ the polynomial is of degree precisely $n$.

Give an example where $k \geqslant 1$ but this polynomial is zero.

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• # Paper 3 , Section I, H

Consider a Markov chain $\left(X_{n}\right)_{n \geqslant 0}$ on a state space $I$.

(a) Define the notion of a communicating class. What does it mean for a communicating class to be closed?

(b) Taking $I=\{1, \ldots, 6\}$, find the communicating classes associated with the transition matrix $P$ given by

$P=\left(\begin{array}{cccccc} 0 & 0 & 0 & 0 & \frac{1}{4} & \frac{3}{4} \\ \frac{1}{4} & 0 & 0 & 0 & \frac{1}{2} & \frac{1}{4} \\ 0 & \frac{1}{2} & 0 & \frac{1}{2} & 0 & 0 \\ 0 & \frac{1}{2} & 0 & 0 & \frac{1}{2} & 0 \\ \frac{1}{4} & \frac{1}{2} & 0 & 0 & 0 & \frac{1}{4} \\ 1 & 0 & 0 & 0 & 0 & 0 \end{array}\right)$

and identify which are closed.

(c) Find the expected time for the Markov chain with transition matrix $P$ above to reach 6 starting from 1 .

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• # Paper 3, Section I, A

Let $f(\theta)$ be a $2 \pi$-periodic function with Fourier expansion

$f(\theta)=\frac{1}{2} a_{0}+\sum_{n=1}^{\infty}\left(a_{n} \cos n \theta+b_{n} \sin n \theta\right)$

Find the Fourier coefficients $a_{n}$ and $b_{n}$ for

f(\theta)=\left\{\begin{aligned} 1, & 0<\theta<\pi \\ -1, & \pi<\theta<2 \pi \end{aligned}\right.

Hence, or otherwise, find the Fourier coefficients $A_{n}$ and $B_{n}$ for the $2 \pi$-periodic function $F$ defined by

$F(\theta)=\left\{\begin{array}{cc} \theta, & 0<\theta<\pi \\ 2 \pi-\theta, & \pi<\theta<2 \pi \end{array}\right.$

$\sum_{r=0}^{\infty} \frac{(-1)^{r}}{2 r+1} \quad \text { and } \quad \sum_{r=0}^{\infty} \frac{1}{(2 r+1)^{2}}$

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• # Paper 3, Section II, A

Let $P(x)$ be a solution of Legendre's equation with eigenvalue $\lambda$,

$\left(1-x^{2}\right) \frac{d^{2} P}{d x^{2}}-2 x \frac{d P}{d x}+\lambda P=0$

such that $P$ and its derivatives $P^{(k)}(x)=d^{k} P / d x^{k}, k=0,1,2, \ldots$, are regular at all points $x$ with $-1 \leqslant x \leqslant 1$.

(a) Show by induction that

$\left(1-x^{2}\right) \frac{d^{2}}{d x^{2}}\left[P^{(k)}\right]-2(k+1) x \frac{d}{d x}\left[P^{(k)}\right]+\lambda_{k} P^{(k)}=0$

for some constant $\lambda_{k}$. Find $\lambda_{k}$ explicitly and show that its value is negative when $k$ is sufficiently large, for a fixed value of $\lambda$.

(b) Write the equation for $P^{(k)}(x)$ in part (a) in self-adjoint form. Hence deduce that if $P^{(k)}(x)$ is not identically zero, then $\lambda_{k} \geqslant 0$.

[Hint: Establish a relation between integrals of the form $\int_{-1}^{1}\left[P^{(k+1)}(x)\right]^{2} f(x) d x$ and $\int_{-1}^{1}\left[P^{(k)}(x)\right]^{2} g(x) d x$ for certain functions $f(x)$ and $\left.g(x) .\right]$

(c) Use the results of parts (a) and (b) to show that if $P(x)$ is a non-zero, regular solution of Legendre's equation on $-1 \leqslant x \leqslant 1$, then $P(x)$ is a polynomial of degree $n$ and $\lambda=n(n+1)$ for some integer $n=0,1,2, \ldots$

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• # Paper 3, Section II, B

The functions $p_{0}, p_{1}, p_{2}, \ldots$ are generated by the formula

$p_{n}(x)=(-1)^{n} x^{-1 / 2} e^{x} \frac{d^{n}}{d x^{n}}\left(x^{n+1 / 2} e^{-x}\right), \quad 0 \leqslant x<\infty$

(a) Show that $p_{n}(x)$ is a monic polynomial of degree $n$. Write down the explicit forms of $p_{0}(x), p_{1}(x), p_{2}(x)$.

(b) Demonstrate the orthogonality of these polynomials with respect to the scalar product

$\langle f, g\rangle=\int_{0}^{\infty} x^{1 / 2} e^{-x} f(x) g(x) d x$

i.e. that $\left\langle p_{n}, p_{m}\right\rangle=0$ for $m \neq n$, and show that

$\left\langle p_{n}, p_{n}\right\rangle=n ! \Gamma\left(n+\frac{3}{2}\right)$

where $\Gamma(y)=\int_{0}^{\infty} x^{y-1} e^{-x} d x$.

(c) Assuming that a three-term recurrence relation in the form

$p_{n+1}(x)=\left(x-\alpha_{n}\right) p_{n}(x)-\beta_{n} p_{n-1}(x), \quad n=1,2, \ldots$

holds, find the explicit expressions for $\alpha_{n}$ and $\beta_{n}$ as functions of $n$.

[Hint: you may use the fact that $\Gamma(y+1)=y \Gamma(y) .]$

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• # Paper 3, Section II, H

Explain what is meant by a two-person zero-sum game with $m \times n$ payoff matrix $A$, and define what is meant by an optimal strategy for each player. What are the relationships between the optimal strategies and the value of the game?

Suppose now that

$A=\left(\begin{array}{cccc} 0 & 1 & 1 & -4 \\ -1 & 0 & 2 & 2 \\ -1 & -2 & 0 & 3 \\ 4 & -2 & -3 & 0 \end{array}\right)$

Show that if strategy $p=\left(p_{1}, p_{2}, p_{3}, p_{4}\right)^{T}$ is optimal for player I, it must also be optimal for player II. What is the value of the game in this case? Justify your answer.

Explain why we must have $(A p)_{i} \leqslant 0$ for all $i$. Hence or otherwise, find the optimal strategy $p$ and prove that it is unique.

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• # Paper 3, Section I, C

The electron in a hydrogen-like atom moves in a spherically symmetric potential $V(r)=-K / r$ where $K$ is a positive constant and $r$ is the radial coordinate of spherical polar coordinates. The two lowest energy spherically symmetric normalised states of the electron are given by

$\chi_{1}(r)=\frac{1}{\sqrt{\pi} a^{3 / 2}} e^{-r / a} \quad \text { and } \quad \chi_{2}(r)=\frac{1}{4 \sqrt{2 \pi} a^{3 / 2}}\left(2-\frac{r}{a}\right) e^{-r / 2 a}$

where $a=\hbar^{2} / m K$ and $m$ is the mass of the electron. For any spherically symmetric function $f(r)$, the Laplacian is given by $\nabla^{2} f=\frac{d^{2} f}{d r^{2}}+\frac{2}{r} \frac{d f}{d r}$.

(i) Suppose that the electron is in the state $\chi(r)=\frac{1}{2} \chi_{1}(r)+\frac{\sqrt{3}}{2} \chi_{2}(r)$ and its energy is measured. Find the expectation value of the result.

(ii) Suppose now that the electron is in state $\chi(r)$ (as above) at time $t=0$. Let $R(t)$ be the expectation value of a measurement of the electron's radial position $r$ at time $t$. Show that the value of $R(t)$ oscillates sinusoidally about a constant level and determine the frequency of the oscillation.

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• # Paper 3, Section II, $18 \mathrm{H}$

Consider the normal linear model $Y=X \beta+\varepsilon$ where $X$ is a known $n \times p$ design matrix with $n-2>p \geqslant 1, \beta \in \mathbb{R}^{p}$ is an unknown vector of parameters, and $\varepsilon \sim N_{n}\left(0, \sigma^{2} I\right)$ is a vector of normal errors with each component having variance $\sigma^{2}>0$. Suppose $X$ has full column rank.

(i) Write down the maximum likelihood estimators, $\hat{\beta}$ and $\hat{\sigma}^{2}$, for $\beta$ and $\sigma^{2}$ respectively. [You need not derive these.]

(ii) Show that $\hat{\beta}$ is independent of $\hat{\sigma}^{2}$.

(iii) Find the distributions of $\hat{\beta}$ and $n \hat{\sigma}^{2} / \sigma^{2}$.

(iv) Consider the following test statistic for testing the null hypothesis $H_{0}: \beta=0$ against the alternative $\beta \neq 0$ :

$T:=\frac{\|\hat{\beta}\|^{2}}{n \hat{\sigma}^{2}} .$

Let $\lambda_{1} \geqslant \lambda_{2} \geqslant \cdots \geqslant \lambda_{p}>0$ be the eigenvalues of $X^{T} X$. Show that under $H_{0}, T$ has the same distribution as

$\frac{\sum_{j=1}^{p} \lambda_{j}^{-1} W_{j}}{Z}$

where $Z \sim \chi_{n-p}^{2}$ and $W_{1}, \ldots, W_{p}$ are independent $\chi_{1}^{2}$ random variables, independent of $Z$.

[Hint: You may use the fact that $X=U D V^{T}$ where $U \in \mathbb{R}^{n \times p}$ has orthonormal columns, $V \in \mathbb{R}^{p \times p}$ is an orthogonal matrix and $D \in \mathbb{R}^{p \times p}$ is a diagonal matrix with $\left.D_{i i}=\sqrt{\lambda_{i}} .\right]$

(v) Find $\mathbb{E} T$ when $\beta \neq 0$. [Hint: If $R \sim \chi_{\nu}^{2}$ with $\nu>2$, then $\mathbb{E}(1 / R)=1 /(\nu-2)$.]

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• # Paper 3, Section I, D

Find the function $y(x)$ that gives a stationary value of the functional

$I[y]=\int_{0}^{1}\left(y^{\prime 2}+y y^{\prime}+y^{\prime}+y^{2}+y x^{2}\right) d x$

subject to the boundary conditions $y(0)=-1$ and $y(1)=e-e^{-1}-\frac{3}{2}$.

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