Update cheatsheet.fr.md

12.temporary_ins/32.sets-systems/30.n3/20.systems/20.overview/cheatsheet.fr.md

@@ -76,7 +76,7 @@ RÉSUMÉ<br>
    \left.\dfrac{dN}{N}\right\lvert_{\,\bigt}=r(t)\,dt`$
 
   $`\begin{align}
-   \left.\dfrac{dN}{dt}\right\lvert_{\,\bigt} &=r\,N(t)\quad \Longrightarrow\quad\left.\dfrac{dN}{N}\right\lvert_{\,\bigt}=r(t)\,dt\\
+   \left.\dfrac{dN}{dt}\right\lvert_{\,\bigt} =r\,N(t)&\quad \Longrightarrow\quad\left.\dfrac{dN}{N}\right\lvert_{\,\bigt}=r(t)\,dt\\
    \\
    &\Longrightarrow\quad\left.\dfrac{dN}{N}\right\lvert_{\,\bigt}=r(t)\,dt
    \end{align}`$
@@ -85,14 +85,17 @@ RÉSUMÉ<br>
 
   <br>
   $`\displaystyle\begin{align}
-   \left.\dfrac{dN}{dt}\right\lvert_{\,\bigt}=r\,N(t)\quad &\Longrightarrow\quad
-   \left.\dfrac{dN}{N}\right\lvert_{\,\bigt}=r(t)\,dt\\
+   \left.\dfrac{dN}{dt}\right\lvert_{\,\bigt}=r\,&N(t)\quad \Longrightarrow\quad
+   \left.\dfrac{dN}{N}\right\lvert_{\,\bigt}=r\,dt\\
    \\
-   &\Longrtightarrow\quad
-   \int_{t_1}^{t_2}\dfrac{dN}{N}=\int_{t_1}^{t_2} r(t)\,dt\\
+   &\Longrightarrow\quad
+   \int_{N(t_1)}^{N(t_2)}\underbrace{\dfrac{dN}{N}_{\begin{array}{c}\text{primitive :}\\ln\,|N|}=\int_{t_1}^{t_2} r\,dt\\
    \\
-   &\Longrtightarrow\quad
-   \int_{t_1}^{t_2}\underbrace{\dfrac{dN}{N}}_{\text{de primitive }ln\,|N|}=\int_{t_1}^{t_2} r(t)\,dt
+   &\Longrightarrow\quad
+   \bigg[ln\,|N|]_{N(t_1)}^{N(t_2)}=\int_{t_1}^{t_2} r \,\bigg[t]_{t_1}^{t_2}\\
+   \\
+   &\Longrightarrow\quad
+   (ln\,|\underbrace{N(t_2)}_{\begin{array}{c}N>0\\|N|=N}end{array}}-\,ln\,|N(t_1)| = r\,(t_2 - t_1)
    \end{align}`$
 
   $`\displaystyle\begin{align}