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>
 
   <br>
   $`\displaystyle\begin{align}
-   \Large{\left.\dfrac{dN}{dt}\right\lvert_{\,\bigt} =r\,N(t)}&\quad \Longrightarrow\quad\left.\dfrac{dN}{N}\right\lvert_{\,\bigt}=r\,dt\\
+   \Large{\left.\dfrac{dN}{dt}\right\lvert_{\,\bigt} =&r\,N(t)}\quad \Longrightarrow\quad\left.\dfrac{dN}{N}\right\lvert_{\,\bigt}=r\,dt\\
    \\
   &\Longrightarrow\quad\int_{N(t_1)}^{N(t_2)}\dfrac{dN}{N}=\int_{t_1}^{t_2} r\,dt\\
    \\
@@ -95,28 +95,29 @@ RÉSUMÉ<br>
    &\Longrightarrow\quad \Large{N(t_2)=N(t_1)\;e^{\,r\,(t_2-t_1)}}
    \end{align}`$
    
-   
-   
-   \\
-   &\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}`$
-
+     <br>
   $`\displaystyle\begin{align}
-   \left.\dfrac{dN}{dt}\right\lvert_{\,\bigt}=r\,N(t)\quad &\Longrightarrow\quad
-   \left.\dfrac{dN}{N}\right\lvert_{\,\bigt}=r(t)\,dt\\
+   \Large{\left.\dfrac{dN}{dt}\right\lvert_{\,\bigt} &=r\,N(t)}\quad \Longrightarrow\quad\left.\dfrac{dN}{N}\right\lvert_{\,\bigt}=r\,dt\\
+   \\
+  &\Longrightarrow\quad\int_{N(t_1)}^{N(t_2)}\dfrac{dN}{N}=\int_{t_1}^{t_2} r\,dt\\
+   \\
+   &\Longrightarrow\quad\big[\,ln\,|N|\,\big]_{N(t_1)}^{N(t_2)}= r \,\big[\,t\,\big]_{t_1}^{t_2}\\
+   \\
+   &\Longrightarrow\quad\underbrace{ln\,|N(t_2)|-\,ln\,|N(t_1)|}_{
+   N>0 \;\Longrightarrow\;|N|\,=\,N} = r\,(t_2 - t_1)\\
    \\
-   &\Longrtightarrow\quad
-   \int_{t_1}^{t_2}\dfrac{dN}{N}=\int_{t_1}^{t_2} r(t)\,dt
+   &\Longrightarrow\quad ln\,N(t_2) = ln\,N(t_1) + r\,(t_2 - t_1)\\
+   \\
+   &\Longrightarrow\quad \underbrace{exp\big[ln\,N(t_2)\big]}_{exp(ln\,x)\;=\;x}
+   =\underbrace{exp\big[ln\,N(t_1) + r\,(t_2 - t_1)\big]}_{exp (a+b)\;=\;exp\,a\times exp\,b}\\
+   \\
+   &\Longrightarrow\quad N(t_2)=N(t_1)\,exp\,\big[r\,(t_2 - t_1)\big]\\
+   \\
+   &\Longrightarrow\quad \Large{N(t_2)=N(t_1)\;e^{\,r\,(t_2-t_1)}}
    \end{align}`$
    
-   $`\displaystyle\begin{align}
-   \left.\dfrac{dN}{dt}\right\lvert_{\,\bigt}=r\,N(t)\quad &\Longrightarrow\quad\left.\dfrac{dN}{N}\right\lvert_{\,\bigt}=r(t)\,dt
-   \end{align}`$
    
-   $`\left.\dfrac{dN}{dt}\right\lvert_{\,\bigt}=r\,N(t)\quad &\Longrightarrow\quad\left.\dfrac{dN}{N}\right\lvert_{\,\bigt}=r(t)\,dt`$
-
-
+