Part A
Below, in an orthogonal coordinate system, are the curves $\mathscr{C}_1$ and $\mathscr{C}_2$, graphical representations of two functions defined and differentiable on $\mathbb{R}$. One of the two functions represented is the derivative of the other. We will denote them $g$ and $g'$. We also specify that:

• The curve $\mathscr{C}_1$ intersects the y-axis at the point with coordinates $(0; 1)$.
• The curve $\mathscr{C}_2$ intersects the y-axis at the point with coordinates $(0; 2)$ and the x-axis at the points with coordinates $(-2; 0)$ and $(1; 0)$.

1. By justifying, associate to each of the functions $g$ and $g'$ its graphical representation.
2. Justify that the equation of the tangent line to the curve representing the function $g$ at the point with x-coordinate 0 is $y = 2x + 1$.

Part B
We consider $(E)$ the differential equation $$y + y' = (2x + 3)\mathrm{e}^{-x}$$ where $y$ is a function of the real variable $x$.

1. Show that the function $f_0$ defined for every real number $x$ by $f_0(x) = (x^2 + 3x)\mathrm{e}^{-x}$ is a particular solution of the differential equation $(E)$.
2. Solve the differential equation $(E_0): y + y' = 0$.
3. Determine the solutions of the differential equation $(E)$.
4. We admit that the function $g$ described in Part A is a solution of the differential equation $(E)$. Then determine the expression of the function $g$.
5. Determine the solutions of the differential equation $(E)$ whose curve has exactly two inflection points.

Part C
We consider the function $f$ defined for every real number $x$ by: $$f(x) = (x^2 + 3x + 2)\mathrm{e}^{-x}$$

1. Prove that the limit of the function $f$ at $+\infty$ is equal to 0.
   We also admit that the limit of the function $f$ at $-\infty$ is equal to $+\infty$.
2. We admit that the function $f$ is differentiable on $\mathbb{R}$. We denote by $f'$ the derivative function of $f$ on $\mathbb{R}$. a. Verify that, for every real number $x$, $f'(x) = (-x^2 - x + 1)\mathrm{e}^{-x}$. b. Determine the sign of the derivative function $f'$ on $\mathbb{R}$ and then deduce the variations of the function $f$ on $\mathbb{R}$.
3. Explain why the function $f$ is positive on the interval $[0; +\infty[$.
4. We will denote by $\mathscr{C}_f$ the curve representing the function $f$ in an orthogonal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath})$. We admit that the function $F$ defined for every real number $x$ by $F(x) = (-x^2 - 5x - 7)\mathrm{e}^{-x}$ is a primitive of the function $f$. Let $\alpha$ be a positive real number. Determine the area $\mathscr{A}(\alpha)$, expressed in square units, of the region of the plane bounded by the x-axis, the curve $\mathscr{C}_f$ and the lines with equations $x = 0$ and $x = \alpha$.

Consider the differential equation
$$(E): \quad y' + 10y = \left(30x^2 + 22x - 8\right)\mathrm{e}^{-5x+1} \quad \text{with} \quad x \in \mathbb{R}$$
where $y$ is a function defined and differentiable on $\mathbb{R}$.

1. Solve on $\mathbb{R}$ the differential equation: $y' + 10y = 0$.
2. Let the function $f$ defined on $\mathbb{R}$ by $$f(x) = \left(6x^2 + 2x - 2\right)\mathrm{e}^{-5x+1}$$ We admit that $f$ is differentiable on $\mathbb{R}$ and we denote $f'$ the derivative function of the function $f$. Justify that $f$ is a particular solution of $(E)$.
3. Give the expression of all solutions of $(E)$.

In a laboratory, a chemical reaction is studied in a closed reactor under certain conditions. The numerical processing of experimental data made it possible to model the evolution of the temperature of this chemical reaction as a function of time. Temperature is expressed in degrees Celsius and time is expressed in minutes. Throughout the exercise, we place ourselves on the time interval $[0;10]$.

Part A

In an orthogonal coordinate system of the plane, we give below the representative curve of the temperature function as a function of time on the interval $[0; 10]$.

1. Determine, by graphical reading, after how much time the temperature returns to its initial value at time $t = 0$.

We call $f$ the temperature function represented by the curve above. We specify that the function $f$ is defined and differentiable on the interval $[0; 10]$. We admit that the function $f$ can be written in the form $f(t) = (at + b)\mathrm{e}^{-0.5t}$ where $a$ and $b$ are two real constants.
2. We admit that the exact value of $f(0)$ is 40. Deduce the value of $b$.
3. We admit that $f$ satisfies the differential equation (E): $y' + 0.5y = 60\mathrm{e}^{-0.5t}$. Determine the value of $a$.

Part B: Study of the function $f$

We admit that the function $f$ is defined for every real $t$ in the interval $[0; 10]$ by $$f(t) = (60t + 40)\mathrm{e}^{-0.5t}$$

1. Show that for every real $t$ in the interval $[0; 10]$, we have: $f'(t) = (40 - 30t)\mathrm{e}^{-0.5t}$.
2. a. Study the direction of variation of the function $f$ on the interval $[0; 10]$. Draw the variation table of the function $f$ showing the images of the values present in the table. b. Show that the equation $f(t) = 40$ has a unique solution $\alpha$ strictly positive on the interval $]0; 10]$. c. Give an approximate value of $\alpha$ to the nearest tenth and give an interpretation in the context of the exercise.
3. We define the average temperature, expressed in degrees Celsius, of this chemical reaction between two times $t_{1}$ and $t_{2}$, expressed in minutes, by $$\frac{1}{t_{2} - t_{1}} \int_{t_{1}}^{t_{2}} f(t)\,\mathrm{dt}$$ a. Using integration by parts, show that $$\int_{0}^{4} f(t)\,\mathrm{dt} = 320 - \frac{800}{\mathrm{e}^{2}}$$ b. Deduce an approximate value, to the nearest degree Celsius, of the average temperature of this chemical reaction during the first 4 minutes.

Part A

We consider the differential equation $$\left(E_1\right): \quad y' + 0.48y = \frac{1}{250}$$ where $y$ is a function of the variable $t$ belonging to the interval $[0; +\infty[$.

1. We consider the constant function $h$ defined on the interval $[0; +\infty[$ by $h(t) = \frac{1}{120}$. Show that the function $h$ is a solution of the differential equation $(E_1)$.
2. Give the general form of the solutions of the differential equation $y' + 0.48y = 0$.
3. Deduce the set of solutions of the differential equation $(E_1)$.

Part B

We are now interested in the evolution of a population of bacteria in a culture medium. At an instant $t = 0$, an initial population of 30000 bacteria is introduced into the medium. We denote $p(t)$ the quantity of bacteria, expressed in thousands of individuals, present in the medium after a time $t$, expressed in hours. We therefore have $p(0) = 30$. We admit that the function $p$ defined on the interval $[0; +\infty[$ is differentiable, strictly positive on this interval and that it is a solution of the differential equation $(E_2)$: $$p' = \frac{1}{250} p \times (120 - p)$$ Let $y$ be the function strictly positive on the interval $[0; +\infty[$ such that, for all $t$ belonging to the interval $[0; +\infty[$, we have $p(t) = \frac{1}{y(t)}$.

1. Show that if $p$ is a solution of the differential equation $(E_2)$, then $y$ is a solution of the differential equation $\left(E_1\right): \quad y' + 0.48y = \frac{1}{250}$.
2. We admit conversely that, if $y$ is a strictly positive solution of the differential equation $(E_1)$, then $p = \frac{1}{y}$ is a solution of the differential equation $(E_2)$. Show that, for all $t$ belonging to the interval $[0; +\infty[$, we have: $$p(t) = \frac{120}{1 + K\mathrm{e}^{-0.48t}} \text{ with } K \text{ a real constant.}$$
3. Using the initial condition, determine the value of $K$.
4. Determine $\lim_{t \rightarrow +\infty} p(t)$. Give an interpretation of this in the context of the exercise.
5. Determine the time required for the bacterial population to exceed 60000 individuals. The result will be given in the form of a rounded value expressed in hours and minutes.

Exercise 1
Part A
Consider the differential equation $$\text{(E)} \quad y' + 0{,}4y = \mathrm{e}^{-0{,}4t}$$ where $y$ is a function of the real variable $t$. We seek the set of functions defined and differentiable on $\mathbb{R}$ that are solutions of this equation.

1. Let $u$ be the function defined on $\mathbb{R}$ by: $u(t) = t\mathrm{e}^{-0{,}4t}$.
   Verify that $u$ is a solution of (E).
   
2. Let $f$ be a function defined and differentiable on $\mathbb{R}$.
   We denote by $g$ the function defined on $\mathbb{R}$ by: $g(t) = f(t) - u(t)$. Let (H) be the differential equation $y' + 0{,}4y = 0$.
   1. [a.] Prove that if the function $g$ is a solution of the differential equation (H) then the function $f$ is a solution of the differential equation (E).
      We will admit that the converse is true.
   2. [b.] Solve the differential equation (H).
   3. [c.] Deduce the solutions of (E).
   4. [d.] Determine the solution $f$ of (E) such that $f(0) = 1$.

Part B
We are interested in blood glucose levels in a person who has just had a meal. The blood glucose level in $\mathrm{g.L}^{-1}$, as a function of time $t$, expressed in hours, elapsed since the end of the meal, is modelled by the function $f$ defined on $[0;6]$ by: $$f(t) = (t+1)\mathrm{e}^{-0{,}4t}$$

1. 1. [a.] Show that, for all $t \in [0;6]$, $f'(t) = (-0{,}4t + 0{,}6)\mathrm{e}^{-0{,}4t}$.
   2. [b.] Study the variations of $f$ on $[0;6]$ then draw up its variation table on this interval.
2. A person is hypoglycaemic when their blood glucose level is below $0{,}7\,\mathrm{g.L}^{-1}$.
   1. [a.] Prove that on the interval $[0;6]$ the equation $f(t) = 0{,}7$ admits a unique solution which we will denote $\alpha$.
   2. [b.] How long after having eaten does this person become hypoglycaemic? Express this time to the nearest minute.
3. We wish to determine the average blood glucose level in $\mathrm{g.L}^{-1}$ in this person during the six hours following the meal.
   1. [a.] Using integration by parts, show that: $$\int_0^6 f(t)\,\mathrm{d}t = -23{,}75\,\mathrm{e}^{-2{,}4} + 8{,}75$$
   2. [b.] Calculate the average blood glucose level in $\mathrm{g.L}^{-1}$ in this person during the six hours following the meal.
   3. [c.] By noting that the function $f$ is a solution of the differential equation (E), explain how we could have obtained this result differently.

Exercise 4
The purpose of this exercise is to study the stopping of a cart on a ride, from the moment it enters the braking zone at the end of the course. We denote $t$ the elapsed time, expressed in seconds, from the moment the cart enters the braking zone. We model the distance travelled by the cart in the braking zone, expressed in metres, as a function of $t$, using a function denoted $d$ defined on $[0; +\infty[$. We thus have $d(0) = 0$. Furthermore, we admit that this function $d$ is differentiable on its domain of definition. We denote $d'$ its derivative function.
Part A
In the figure (Fig. 2), we have drawn in an orthonormal coordinate system:

• the representative curve $\mathscr{C}_d$ of the function $d$;
• the tangent $T$ to the curve $\mathscr{C}_d$ at point A with abscissa 4.7;
• the asymptote $\Delta$ to $\mathscr{C}_d$ at $+\infty$.

In this part, no justification is expected. With the precision that the graph allows, answer the questions below. According to this model:

1. After how much time will the cart have travelled 15 m in the braking zone?
2. What minimum length must be provided for the braking zone?
3. What is the value of $d'(4{,}7)$? Interpret this result in the context of the exercise.

Part B
We recall that $t$ denotes the elapsed time, in seconds, from the moment the cart enters the braking zone. We model the instantaneous velocity of the cart, in metres per second ($\mathrm{m.s^{-1}}$), as a function of $t$, by a function $v$ defined on $[0; +\infty[$. We admit that:

• the function $v$ is differentiable on its domain of definition, and we denote $v'$ its derivative function;
• the function $v$ is a solution of the differential equation $$(E): \quad y' + 0{,}6\, y = \mathrm{e}^{-0{,}6t},$$ where $y$ is an unknown function and $y'$ is the derivative function of $y$.

We further specify that, upon arrival in the braking zone, the velocity of the cart is equal to $12\,\mathrm{m.s^{-1}}$, that is $v(0) = 12$.

1. 1. [a.] We consider the differential equation $$(E'): \quad y' + 0{,}6\, y = 0$$ Determine the solutions of the differential equation $(E')$ on $[0; +\infty[$.
   2. [b.] Let $g$ be the function defined on $[0; +\infty[$ by $g(t) = t\,\mathrm{e}^{-0{,}6t}$. Verify that the function $g$ is a solution of the differential equation $(E)$.
   3. [c.] Deduce the solutions of the differential equation $(E)$ on $[0; +\infty[$.
   4. [d.] Deduce that for every real $t$ belonging to the interval $[0; +\infty[$, we have: $$v(t) = (12 + t)\,\mathrm{e}^{-0{,}6t}$$
2. In this question, we study the function $v$ on $[0; +\infty[$.
   1. [a.] Show that for every real $t \in [0; +\infty[$, $v'(t) = (-6{,}2 - 0{,}6t)\,\mathrm{e}^{-0{,}6t}$.
   2. [b.] By admitting that: $$v(t) = 12\,\mathrm{e}^{-0{,}6t} + \frac{1}{0{,}6} \times \frac{0{,}6t}{\mathrm{e}^{0{,}6t}}$$ determine the limit of $v$ at $+\infty$.
   3. [c.] Study the direction of variation of the function $v$ and draw up its complete variation table. Justify.
   4. [d.] Show that the equation $v(t) = 1$ has a unique solution $\alpha$, of which you will give an approximate value to the nearest tenth.
3. When the velocity of the cart is less than or equal to 1 metre per second, a mechanical system is triggered allowing its complete stopping. Determine after how much time this system comes into action. Justify.

Part C
We recall that for every real $t$ belonging to the interval $[0; +\infty[$: $$v(t) = (12 + t)\,\mathrm{e}^{-0{,}6t}.$$ We admit that for every real $t$ in the interval $[0; +\infty[$: $$d(t) = \int_0^t v(x)\,\mathrm{d}x$$

1. Using integration by parts, show that the distance travelled by the cart between times 0 and $t$ is given by: $$d(t) = \mathrm{e}^{-0{,}6t}\left(-\frac{5}{3}t - \frac{205}{9}\right) + \frac{205}{9}$$
2. We recall that the stopping device is triggered when the velocity of the cart is less than or equal to 1 metre per second. Determine, according to this model, an approximate value to the nearest hundredth of the distance travelled by the cart in the braking zone before the triggering of this device.

For each of the following statements, indicate whether it is true or false. Each answer must be justified. An unjustified answer earns no points. In this exercise, the questions are independent of one another.

1. We consider the differential equation: $$(E) \quad y' = \frac{1}{2}y + 4.$$ Statement 1: The solutions of $(E)$ are the functions $f$ defined on $\mathbb{R}$ by: $$f(x) = k\mathrm{e}^{\frac{1}{2}x} - 8, \quad \text{with } k \in \mathbb{R}.$$
   
2. In a final year class, there are 18 girls and 14 boys. A volleyball team is formed by randomly choosing 3 girls and 3 boys. Statement 2: There are 297024 possibilities for forming such a team.
   
3. Let $(v_n)$ be the sequence defined for every natural integer $n$ by: $$v_n = \frac{n}{2 + \cos(n)}.$$ Statement 3: The sequence $(v_n)$ diverges to $+\infty$.
   
4. In space with respect to an orthonormal coordinate system $(\mathrm{O}; \vec{\imath}, \vec{\jmath}, \vec{k})$, we consider the points $\mathrm{A}(1; 1; 2)$, $\mathrm{B}(5; -1; 8)$ and $\mathrm{C}(2; 1; 3)$. Statement 4: $\overrightarrow{\mathrm{AB}} \cdot \overrightarrow{\mathrm{AC}} = 10$ and a measure of the angle $\widehat{\mathrm{BAC}}$ is $30^\circ$.
   
5. We consider a function $h$ defined on $]0; +\infty[$ whose second derivative is defined on $]0; +\infty[$ by: $$h''(x) = x\ln x - 3x.$$ Statement 5: The function $h$ is convex on $[\mathrm{e}^3; +\infty[$.

For each of the following statements, indicate whether it is true or false. Each answer must be justified. An unjustified answer earns no points.

1. Let $E$ and $F$ be the sets $E = \{1; 2; 3; 4; 5; 6; 7\}$ and $F = \{0; 1; 2; 3; 4; 5; 6; 7; 8; 9\}$. Statement $\mathbf{n^\circ 1}$: There are more 3-tuples of distinct elements of $E$ than 4-element combinations of $F$.
2. In the orthonormal coordinate system, we have represented the square function, denoted $f$, as well as the square ABCD with side 3. Statement $\mathbf{n^\circ 2}$: The shaded region and the square ABCD have the same area.
3. We consider the integral $J$ below: $$J = \int_1^2 x\ln(x)\,\mathrm{d}x$$ Statement $\mathbf{n^\circ 3}$: Integration by parts makes it possible to obtain: $J = \dfrac{7}{11}$.
4. On $\mathbb{R}$, we consider the differential equation $$(E): \quad y' = 2y - \mathrm{e}^x.$$ Statement $\mathbf{n^\circ 4}$: The function $f$ defined on $\mathbb{R}$ by $f(x) = \mathrm{e}^x + \mathrm{e}^{2x}$ is a solution of the differential equation $(E)$.
5. Let $x$ be given in $[0; 1[$. We consider the sequence $(u_n)$ defined for any natural integer $n$ by: $$u_n = (x-1)\mathrm{e}^n + \cos(n).$$ Statement $\mathbf{n^\circ 5}$: The sequence $(u_n)$ diverges to $-\infty$.

$\mathcal{P}$ 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 $U(P)(t) = e^t D\left(te^{-t}P^{\prime}(t)\right)$.
We consider on $[0, +\infty[$ the differential equation $$(E_n): \quad tP^{\prime\prime} + (1-t)P^{\prime} + nP = 0$$ with $n \in \mathbb{N}$ and unknown $P \in \mathcal{P}$.
VIII.G.1) By applying the transformation $L$ with $\lambda(t) = t$ to $(E_n)$, show that if $P$ is a solution of $(E_n)$ on $[0, +\infty[$, then its image $Q$ by $L$ is a solution of a differential equation $(E_n^{\prime})$ of order 1 on $]1, +\infty[$.
VIII.G.2) Solve the equation $(E_n^{\prime})$ on $]1, +\infty[$ and deduce from this the eigenvalues and eigenvectors of the endomorphism $U$.
VIII.G.3) What is the relationship between the above and the polynomial functions defined for $n \in \mathbb{N}$ by $P_n(t) = e^t D^n\left(e^{-t}t^n\right)$?

We study the asymptotic behavior near $+ \infty$ of a solution $z \in E$ of the differential equation defined on $] 0 , + \infty [$, with $\lambda \in \mathbb { R } ^ { * }$ :
$$z ^ { \prime \prime } + \left( 1 + \frac { \lambda } { x ^ { 2 } } \right) z = 0 \tag{III.3}$$
Let $x _ { 0 }$ in $] 0 , + \infty [$.
By considering the differential equation (III.3) in the form $z ^ { \prime \prime } + z = g$ with $g ( x ) = \frac { - \lambda } { x ^ { 2 } } z ( x )$, solve it on $] 0 , + \infty [$ using the method of variation of constants. Deduce that there exist two real numbers $A$ and $B$ such that
$$\forall x \in ] 0 , + \infty \left[ \quad z ( x ) = A \cos ( x ) + B \sin ( x ) + \lambda \int _ { x _ { 0 } } ^ { x } z ( u ) \sin ( u - x ) \frac { \mathrm { d } u } { u ^ { 2 } } \right.$$

To each function $f \in E$, we associate the function $U ( f )$ with derivative $U(f)^\prime(x) = \mathrm { e } ^ { x } \int _ { x } ^ { + \infty } f ( t ) \frac { \mathrm { e } ^ { - t } } { t } \mathrm {~d} t$. Let $f \in E$. Show that $U ( f )$ is of class $\mathcal { C } ^ { 2 }$ on $\mathbb { R } _ { + } ^ { * }$ and that the function $U ( f )$ is a solution on $\mathbb { R } _ { + } ^ { * }$ of the differential equation $$y ^ { \prime \prime } - y ^ { \prime } = - \frac { f ( x ) } { x }$$

Problem 1
I. Find the general solution of the following differential equation:
$$\frac { d ^ { 4 } y } { d x ^ { 4 } } - 2 \cdot \frac { d ^ { 3 } y } { d x ^ { 3 } } + 2 \frac { d y } { d x } - y = 9 e ^ { - 2 x }$$
Here, $e$ denotes the base of the natural logarithm.
II. Find the value of the following integral:
$$\int _ { 0 } ^ { 1 } x ^ { m } ( \log x ) ^ { n } d x$$
Here, $m$ and $n$ are non-negative integers.
III. We define $I ( m )$ as
$$I ( m ) \equiv \int _ { 0 } ^ { 1 } x ^ { m } \arccos x \, d x$$
Here, $m$ is a non-negative integer. Use the principal values of inverse trigonometric functions.

1. Find the value of $I ( 0 )$.
2. Find the value of $I ( 1 )$.
3. Express $I ( m )$ in terms of $m$ and $I ( m - 2 )$ when $m \geq 2$.
4. Find the value of $I ( m )$.

Problem 1

I. Find the value of the following definite integral:
$$I = \int _ { 2 } ^ { 4 } \frac { d x } { \sqrt { ( x - 2 ) ( 4 - x ) } }$$
II. Find the general solution and the singular solution of the following differential equation:
$$y = x \frac { d y } { d x } + \frac { d y } { d x } + \left( \frac { d y } { d x } \right) ^ { 2 }$$
III. Find the general solution of the following differential equation:
$$x ^ { 2 } \frac { d ^ { 2 } y } { d x ^ { 2 } } - x \frac { d y } { d x } - 8 y = x ^ { 2 }$$

I. Find the general solutions of the following differential equations.

1. $\frac { d ^ { 2 } y } { d x ^ { 2 } } + 2 \frac { d y } { d x } - 3 y = e ^ { x } \cos x$
2. $\frac { d ^ { 2 } y } { d x ^ { 2 } } + \frac { 1 } { x } \frac { d y } { d x } + \frac { 4 } { x ^ { 2 } } y = \left( \frac { 2 \log x } { x } \right) ^ { 2 }$

II. Answer the following questions for the partial differential equation represented in Equation (3) and the boundary conditions represented in Equations (4)-(7):
$$\begin{aligned} & \frac { \partial ^ { 2 } u ( x , y ) } { \partial x ^ { 2 } } + \frac { \partial ^ { 2 } u ( x , y ) } { \partial y ^ { 2 } } = 0 \quad ( 0 \leq x , 0 \leq y \leq 1 ) \\ & \left\{ \begin{array} { l } \lim _ { x \rightarrow + \infty } u ( x , y ) = 0 \\ \left. \frac { \partial u ( x , y ) } { \partial y } \right| _ { y = 0 } = 0 \\ u ( x , 1 ) = 0 \\ \left. \frac { \partial u ( x , y ) } { \partial x } \right| _ { x = 0 } = 1 + \cos \pi y \end{array} \right. \end{aligned}$$

1. Find the solution which satisfies Equations (3) and (4) in the form of $u ( x , y ) = X ( x ) \cdot Y ( y )$.
2. Find the solution satisfying Equations (5) and (6) for the solution of Question II.1.
3. Find the solution of the partial differential equation (3) satisfying all the boundary conditions given in Equations (4)-(7), using the solution of Question II.2.

Problem 1
I. Obtain the general solution of the following differential equation: $$x ^ { 2 } \frac { d ^ { 2 } y } { d x ^ { 2 } } - x \frac { d y } { d x } + y = x ^ { 3 }$$
II. Obtain the general solution of the following differential equation: $$x ^ { 2 } \frac { d y } { d x } - x ^ { 2 } y ^ { 2 } + x y + 1 = 0$$ Note that $y = \frac { 1 } { x }$ is a particular solution.
III. Let $I _ { n }$ be defined by: $$I _ { n } = \int _ { 0 } ^ { \frac { \pi } { 4 } } \tan ^ { n } x \, d x$$ where $n$ is a non-negative integer.

1. Calculate $I _ { 0 } , I _ { 1 }$, and $I _ { 2 }$.
2. Calculate $I _ { n }$ for $n \geq 2$.

Answer all the following questions.
I. Find the following limit value:
$$\lim _ { x \rightarrow 0 } \frac { b ^ { x } - c ^ { x } } { a x } \quad ( a , b , c > 0 )$$
II. Find the general solutions of the following differential equations.
$$\begin{aligned} & \text { 1. } \frac { \mathrm { d } y } { \mathrm {~d} x } - \frac { y } { x } = \log x \quad ( x > 0 ) \\ & \text { 2. } \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - \frac { \mathrm { d } y } { \mathrm {~d} x } - 2 y = 2 x ^ { 2 } + 2 x \end{aligned}$$
III. Let $a _ { n }$ be defined by
$$a _ { n } = \frac { n ! } { n ^ { n + \frac { 1 } { 2 } } e ^ { - n } }$$
where $n$ is a positive integer and $e$ is the base of natural logarithm. Find $\lim _ { n \rightarrow \infty } \frac { a _ { n } } { a _ { n + 1 } }$. Note that the function $y = x ^ { - 1 } ( x > 0 )$ is convex downward.

Let $t$ be a real independent variable, and let $x ( t )$ and $y ( t )$ be real-valued functions. Answer the following questions.
(1) Find all solutions of the following ordinary differential equation
$$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 \frac { \mathrm {~d} x } { \mathrm {~d} t } + x = \cos ( t )$$
which are bounded when $t \rightarrow - \infty$.
(2) Find all solutions $x ( t )$ and $y ( t )$ of the following ordinary differential equations
$$\begin{aligned} & \frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 \frac { \mathrm {~d} x } { \mathrm {~d} t } + x - y = \cos ( t ) \\ & \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } + 2 \frac { \mathrm {~d} y } { \mathrm {~d} t } + y - x = 0 \end{aligned}$$
which are bounded when $t \rightarrow - \infty$.
(3) By converting the following ordinary differential equation
$$e ^ { - t } x ^ { 2 } - 2 \frac { \mathrm {~d} x } { \mathrm {~d} t } + x = 0$$
to a linear ordinary differential equation with an appropriate change of variable, find the solution $x ( t )$ that satisfies $x ( 0 ) = \frac { 1 } { 2 }$.

Problem 1

I. Find the general solutions $y ( x )$ for the following differential equations:

1. $\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { y } { x } = \left( \frac { y } { x } \right) ^ { 3 }$,
2. $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 5 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 6 [ y + \cos ( 3 x ) ] = 0$.

II. Consider the curve $C$ given by the following polar equation in the polar coordinate system $( r , \theta )$ with the origin $O$ on the $x y$-orthogonal coordinate plane as the pole, and the positive part of the $x$-axis as the starting line:
$$r = 2 + \cos \theta \quad ( 0 \leq \theta < 2 \pi )$$

1. Calculate the area of the region enclosed by the curve $C$.
2. Consider the tangent line at the point $( r , \theta ) = \left( \frac { 4 + \sqrt { 2 } } { 2 } , \frac { \pi } { 4 } \right)$ on the curve $C$. Find the slope of this tangent line in the $x y$-orthogonal coordinate system.