diff --git a/analyse/main.tex b/analyse/main.tex index ef26b64..b9a42bc 100644 --- a/analyse/main.tex +++ b/analyse/main.tex @@ -40,7 +40,7 @@ \paragraph{Dérivées et Primitives} - Usuelles~: + \begin{multicols}{2} \begin{tabularx}{\linewidth}{YY} \toprule @@ -82,48 +82,46 @@ \bottomrule \end{tabularx} - Composées~: + \columnbreak - \begin{multicols}{2} + \begin{tabularx}{\linewidth}{lY} + \toprule + \multirow{2}{*}{Linéarité} & $(u + v)' = u' + v'$ \\ + & $(au)' = au'$ \\ + \midrule + Produit & $(uv)' = u'v + uv'$ \\ + \midrule + Inverse & $\left(\frac{1}{v}\right)' = - \frac{v'}{v^2}$ \\ + \midrule + Quotient & $\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}$ \\ + \midrule + Composée & $(f(u))' = u'f'(u)$ \\ + \bottomrule + \end{tabularx} - \begin{tabularx}{\linewidth}{lY} - \toprule - \multirow{2}{*}{Linéarité} & $(u + v)' = u' + v'$ \\ - & $(au)' = au'$ \\ - \midrule - Produit & $(uv)' = u'v + uv'$ \\ - \midrule - Inverse & $\left(\frac{1}{v}\right)' = - \frac{v'}{v^2}$ \\ - \midrule - Quotient & $\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}$ \\ - \midrule - Composée & $(f(u))' = u'f'(u)$ \\ - \bottomrule - \end{tabularx} + \begin{tabularx}{\linewidth}{YY} + \toprule + Fonction & Primitive \\ + \toprule + $u'u^n$ & $\frac{u^{n+1}}{n+1}$ \\ + \midrule + $\frac{u'}{u^2}$ & $-\frac{1}{u}$ \\ + \midrule + $\frac{u'}{\sqrt{u}}$ & $2\sqrt{u}$ \\ + \midrule + $u'\cos{u}$ & $\sin{u}$ \\ + \midrule + $u'\sin{u}$ & $-\cos{u}$ \\ + \midrule + $\frac{u'}{u}$ & $\ln{|u|}$ \\ + \midrule + $u'e^u$ & $e^u$ \\ + \midrule + $\frac{u'}{1 + u^2}$ & $\arctan{u}$ \\ + \bottomrule + \end{tabularx} - \begin{tabularx}{\linewidth}{YY} - \toprule - Fonction & Primitive \\ - \toprule - $u'u^n$ & $\frac{u^{n+1}}{n+1}$ \\ - \midrule - $\frac{u'}{u^2}$ & $-\frac{1}{u}$ \\ - \midrule - $\frac{u'}{\sqrt{u}}$ & $2\sqrt{u}$ \\ - \midrule - $u'\cos{u}$ & $\sin{u}$ \\ - \midrule - $u'\sin{u}$ & $-\cos{u}$ \\ - \midrule - $\frac{u'}{u}$ & $\ln{|u|}$ \\ - \midrule - $u'e^u$ & $e^u$ \\ - \midrule - $\frac{u'}{1 + u^2}$ & $\arctan{u}$ \\ - \bottomrule - \end{tabularx} - - \end{multicols} + \end{multicols} \paragraph{Intégrales} $\int_a^b f(x)\dif x = [F(x)]_a^b = F(b) - F(a)$ @@ -136,7 +134,6 @@ \bottomrule \end{tabularx} - \paragraph{Équations différentielles} \begin{tabularx}{\linewidth}{lllc} @@ -147,11 +144,26 @@ \midrule \multicolumn{2}{l}{$ay' + by = f(x)$} & $y_0 + \lambda e^{rx} \quad \text{ avec } r = \frac{-b}{a}$ & \makecell{$y_0$ solution particulière de \\ $ay' + by = f(x)$ \\ $f$ une fonction et $a, b, \lambda\in\mathbb{R}$} \\ \midrule - \multirow{3}{*}{$ay'' + by' + cy = 0$} & $\Delta > 0$ & $\lambda e^{r_1 x} + \mu e^{r_2 x} + y_1$ & \multirowcell{3}[0pt][c]{$\lambda, \mu \in \mathbb{R}$, $y_1$ solution particulière \\ $\alpha = \frac{-b}{2a} \quad \beta = \frac{\sqrt{|\Delta|}}{2a}$} \\ + \multirow{3}{*}{$ay'' + by' + cy = 0$} & $\Delta > 0$ & $\lambda e^{r_1 x} + \mu e^{r_2 x}$ & \multirowcell{3}[0pt][c]{$\lambda, \mu \in \mathbb{R}$ \\ $\alpha = \frac{-b}{2a} \quad \beta = \frac{\sqrt{|\Delta|}}{2a}$} \\ \cline{2-3} - & $\Delta = 0$ & $(\lambda x + \mu) e^{r_0 x} + y_1$ & \\ + & $\Delta = 0$ & $(\lambda x + \mu) e^{r_0 x}$ & \\ \cline{2-3} - & $\Delta < 0$ & $e^{\alpha x}(\lambda\cos{(\beta x)} + \mu\sin{(\beta x)}) + y_1$ & \\ + & $\Delta < 0$ & $e^{\alpha x}(\lambda\cos{(\beta x)} + \mu\sin{(\beta x)})$ & \\ + \bottomrule + \end{tabularx} + + \paragraph{Solutions particulières des équations différentielles de 2\up{nd} ordre} + + \begin{tabularx}{\linewidth}{XX} + \toprule + \multicolumn{2}{l}{Second membre du type $e^{\alpha x}P(x)$} \\ + $\alpha$ non racine & $y_1 = e^{\alpha x} Q(x)$ \\ + $\alpha$ racine simple & $y_1 = x e^{\alpha x} Q(x)$ \\ + $\alpha$ racine double & $y_1 = x^2 e^{\alpha x} Q(x)$ \\ + \midrule + \multicolumn{2}{l}{Second membre du type $e^{\alpha x}(P_1(x)\cos(\beta x) + P_2(x)\sin(\beta x))$} \\ + $\alpha + i\beta$ non racine & $y_1 = e^{\alpha x}(Q_1(x)\cos(\beta x) + Q_2(x)\sin(\beta x))$ \\ + $\alpha + i\beta$ racine & $y_1 = x e^{\alpha x}(Q_1(x)\cos(\beta x) + Q_2(x)\sin(\beta x))$ \\ \bottomrule \end{tabularx} diff --git a/cours.sty b/cours.sty index a94a6c0..4e4067e 100644 --- a/cours.sty +++ b/cours.sty @@ -2,7 +2,7 @@ \usepackage[ %showframe, - a4paper,includeheadfoot,margin=2cm + a4paper,includeheadfoot,margin=2cm,top=1cm,bottom=1cm ]{geometry} \setcounter{tocdepth}{2}