6. The Maclaurin series for the function $f$ is given by $f ( x ) = \sum _ { n = 2 } ^ { \infty } \frac { ( - 1 ) ^ { n } ( 2 x ) ^ { n } } { n - 1 }$ on its interval of convergence.
(a) Find the interval of convergence for the Maclaurin series of $f$. Justify your answer.
(b) Show that $y = f ( x )$ is a solution to the differential equation $x y ^ { \prime } - y = \frac { 4 x ^ { 2 } } { 1 + 2 x }$ for $| x | < R$, where $R$ is the radius of convergence from part (a).

WRITE ALL WORK IN THE EXAM BOOKLET. END OF EXAM

For a quadratic function $f ( x )$, the function $g ( x ) = f ( x ) e ^ { - x }$ satisfies the following conditions. (가) The points $( 1 , g ( 1 ) )$ and $( 4 , g ( 4 ) )$ are inflection points of the curve $y = g ( x )$. (나) The number of tangent lines drawn from the point $( 0 , k )$ to the curve $y = g ( x )$ is 3 when $k$ is in the range $- 1 < k < 0$. Find the value of $g ( - 2 ) \times g ( 4 )$. [4 points]

$\mathcal{P}$ denotes the vector space of polynomial functions with complex coefficients, with inner product $\langle P, Q \rangle = \displaystyle\int_0^{+\infty} e^{-t}\bar{P}(t)Q(t)\,dt$, and $U$ is the endomorphism of $\mathcal{P}$ defined by $U(P)(t) = e^t D\left(te^{-t}P^{\prime}(t)\right)$.
Let $\lambda$ be an eigenvalue of $U$ and $P$ an eigenvector associated with it.
VIII.F.1) Show that $P$ is a solution of a simple linear differential equation that we will specify.
VIII.F.2) What is the relationship between $\lambda$ and the degree of $P$?

Let $n$ in $\mathbb { N } ^ { * }$, verify that for real $x$
$$\frac { \mathrm { d } } { \mathrm {~d} x } \left( x ^ { n } \varphi _ { n } ( x ) \right) = x ^ { n } \varphi _ { n - 1 } ( x )$$

We study the differential equation with unknown $y$
$$x ^ { 2 } y ^ { \prime \prime } + x y ^ { \prime } + \left( x ^ { 2 } - n ^ { 2 } \right) y = 0 \tag{III.1}$$
We seek solutions in the set $E = \mathcal { C } ^ { 2 } ( ] 0 , + \infty [ )$.
Using the power series development of $\varphi _ { n }$ (II.C.3), show that $\varphi _ { n }$ is a solution on $[ 0 , + \infty [$ of (III.1).

For $p \in \mathbb { N } ^ { * }$, consider a polynomial $P \in \mathbb { R } [ X ]$ such that the polynomial function $x \mapsto P ( x )$ is a solution of the equation $\left( E _ { p } \right) : x \left( y ^ { \prime \prime } - y ^ { \prime } \right) + p y = 0$. For all $x \in \mathbb { R }$, we denote $h ( x ) = \mathrm { e } ^ { - x } P ( x )$. Show that the function $h$ is a solution of the differential equation $x \left( y ^ { \prime \prime } + y ^ { \prime } \right) + p y = 0$ on $\mathbb { R } _ { + } ^ { * }$.

129. The function with the rule $f(x) = \displaystyle\lim_{n \to +\infty} \left(1 - \dfrac{3x}{n}\right)^n$ is defined for every real number $x$. Which statement is correct?

• [(1)] $f''(x) + 6f'(x) + 9f(x) = 0$
• [(2)] $f''(x) + 3f'(x) + 2f(x) = 0$
• [(3)] $f''(x) - 6f'(x) + 9f(x) = 0$
• [(4)] $f''(x) - 3f'(x) + 2f(x) = 0$

10. Determine which are the values of the parameter $k \in \mathbb{R}$ for which the function $y(x) = 2e^{kx+2}$ is a solution of the differential equation $y'' - 2y' - 3y = 0$.
\footnotetext{Maximum duration of the examination: 6 hours. The use of scientific and/or graphical calculators is permitted provided they are not equipped with symbolic calculation capacity (O.M. no. 350 Art. 18 paragraph 8). The use of a bilingual dictionary (Italian–language of the country of origin) is permitted for candidates whose native language is not Italian. It is not permitted to leave the Institute before 3 hours have elapsed from the dictation of the theme.}