Main topics covered: Differential equations; exponential function.

We consider the differential equation

$$\text { (E) } y ^ { \prime } = y + 2 x \mathrm { e } ^ { x }$$

We seek the set of functions defined and differentiable on the set $\mathbb { R }$ of real numbers that are solutions to this equation.

\begin{enumerate}
  \item Let $u$ be the function defined on $\mathbb { R }$ by $u ( x ) = x ^ { 2 } \mathrm { e } ^ { x }$. We admit that $u$ is differentiable and we denote $u ^ { \prime }$ its derivative function. Prove that $u$ is a particular solution of $( E )$.
  \item Let $f$ be a function defined and differentiable on $\mathbb { R }$. We denote $g$ the function defined on $\mathbb { R }$ by:
$$g ( x ) = f ( x ) - u ( x )$$
a. Prove that if the function $f$ is a solution of the differential equation $( E )$ then the function $g$ is a solution of the differential equation: $y ^ { \prime } = y$.\\
We admit that the converse of this property is also true.\\
b. Using the solution of the differential equation $y ^ { \prime } = y$, solve the differential equation (E).
  \item Study of the function $u$\\
a. Study the sign of $u ^ { \prime } ( x )$ for $x$ varying in $\mathbb { R }$.\\
b. Draw the table of variations of the function $u$ on $\mathbb { R }$ (limits are not required).\\
c. Determine the largest interval on which the function $u$ is concave.
\end{enumerate}