diff --git a/TP Physique Equation de la chaleur/TP Physique Equation de la chaleur.ipynb b/TP Physique Equation de la chaleur/TP Physique Equation de la chaleur.ipynb
index 4d9d773..d44d0bc 100644
--- a/TP Physique Equation de la chaleur/TP Physique Equation de la chaleur.ipynb	
+++ b/TP Physique Equation de la chaleur/TP Physique Equation de la chaleur.ipynb	
@@ -43,20 +43,19 @@
     "\n",
     "Système linéaire\n",
     "$$\n",
-    "-2T0 + T(1) = -T0\n",
-    "\n",
-    "T0 - 2T1 + T2 = 0\n",
-    "\n",
-    "T1 - 2T2 + T3 = 0\n",
-    "\n",
-    ".\n",
-    ".\n",
-    ".\n",
-    "\n",
-    "TN - 2T(N+1) = -T1+\n",
+    "\\begin{cases}\n",
+    "-2T_0 + T_1 = -T_0 \\\\\n",
+    "T_0 - 2T_1 + T_2 = 0 \\\\\n",
+    "T_1 - 2T_2 + T_3 = 0 \\\\\n",
+    "\\vdots \\\\\n",
+    "T_{N-1} - 2T_N + T_{N+1} = 0 \\\\\n",
+    "T_N - 2T_{N+1} = -T_1\n",
+    "\\end{cases}\n",
     "$$\n",
+    "\n",
     "=> MT = b\n",
     "\n",
+    "### Question 3: Montrer que l'équation discrétisée peut s'écrire: MT = b, avc M une matrice de taille N X N et b un vecteur de taille N. On précisera les termes non nuls de M et b.\n",
     "Matrice du système (simplifiée pour N=6, normalement N indéfini)\n",
     "$$\n",
     "\\begin{bmatrix}\n",
@@ -87,8 +86,104 @@
     "0\\\\\n",
     "-T1+\n",
     "\\end{bmatrix}\n",
-    "$$"
+    "$$\n",
+    "\n",
+    "### Question 4: Sous python, construire la matrice M et le vecteur b à l'aide de numpy. Résoudre ensuite le système linéaire à l'aide de la fonction numpy.linalg.solve(M, b).\n"
    ]
+  },
+  {
+   "metadata": {
+    "ExecuteTime": {
+     "end_time": "2025-04-01T07:53:00.484017Z",
+     "start_time": "2025-04-01T07:53:00.474195Z"
+    }
+   },
+   "cell_type": "code",
+   "source": [
+    "import numpy as np\n",
+    "\n",
+    "# Constantes et paramètres\n",
+    "N = 100\n",
+    "\n",
+    "M = np.zeros((N + 1, N + 1))\n",
+    "\n",
+    "b = np.zeros(N + 1)\n",
+    "\n",
+    "# Conditions aux bords\n",
+    "T0 = 2\n",
+    "T1 = 25\n",
+    "\n",
+    "# Remplissage de la matrice M\n",
+    "for i in range(N + 1):\n",
+    "    if i == 0:\n",
+    "        M[i, i] = -2\n",
+    "        M[i, i + 1] = 1\n",
+    "    elif i == N:\n",
+    "        M[i, i - 1] = 1\n",
+    "        M[i, i] = -2\n",
+    "    else:\n",
+    "        M[i, i - 1] = 1\n",
+    "        M[i, i] = -2\n",
+    "        M[i, i + 1] = 1\n",
+    "\n",
+    "# Affichage de la matrice M\n",
+    "print(M)\n",
+    "\n",
+    "# Remplissage du vecteur b\n",
+    "b[0] = -T0\n",
+    "b[N] = -T1\n",
+    "\n",
+    "# Affichage du vecteur b\n",
+    "print(b)\n",
+    "\n",
+    "# Résolution du système linéaire\n",
+    "T = np.linalg.solve(M, b)\n",
+    "\n",
+    "# Affichage de la solution\n",
+    "print(T)"
+   ],
+   "id": "d480d6e691bf7fbf",
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "[[-2.  1.  0. ...  0.  0.  0.]\n",
+      " [ 1. -2.  1. ...  0.  0.  0.]\n",
+      " [ 0.  1. -2. ...  0.  0.  0.]\n",
+      " ...\n",
+      " [ 0.  0.  0. ... -2.  1.  0.]\n",
+      " [ 0.  0.  0. ...  1. -2.  1.]\n",
+      " [ 0.  0.  0. ...  0.  1. -2.]]\n",
+      "[ -2.   0.   0.   0.   0.   0.   0.   0.   0.   0.   0.   0.   0.   0.\n",
+      "   0.   0.   0.   0.   0.   0.   0.   0.   0.   0.   0.   0.   0.   0.\n",
+      "   0.   0.   0.   0.   0.   0.   0.   0.   0.   0.   0.   0.   0.   0.\n",
+      "   0.   0.   0.   0.   0.   0.   0.   0.   0.   0.   0.   0.   0.   0.\n",
+      "   0.   0.   0.   0.   0.   0.   0.   0.   0.   0.   0.   0.   0.   0.\n",
+      "   0.   0.   0.   0.   0.   0.   0.   0.   0.   0.   0.   0.   0.   0.\n",
+      "   0.   0.   0.   0.   0.   0.   0.   0.   0.   0.   0.   0.   0.   0.\n",
+      "   0.   0. -25.]\n",
+      "[ 2.2254902   2.45098039  2.67647059  2.90196078  3.12745098  3.35294118\n",
+      "  3.57843137  3.80392157  4.02941176  4.25490196  4.48039216  4.70588235\n",
+      "  4.93137255  5.15686275  5.38235294  5.60784314  5.83333333  6.05882353\n",
+      "  6.28431373  6.50980392  6.73529412  6.96078431  7.18627451  7.41176471\n",
+      "  7.6372549   7.8627451   8.08823529  8.31372549  8.53921569  8.76470588\n",
+      "  8.99019608  9.21568627  9.44117647  9.66666667  9.89215686 10.11764706\n",
+      " 10.34313725 10.56862745 10.79411765 11.01960784 11.24509804 11.47058824\n",
+      " 11.69607843 11.92156863 12.14705882 12.37254902 12.59803922 12.82352941\n",
+      " 13.04901961 13.2745098  13.5        13.7254902  13.95098039 14.17647059\n",
+      " 14.40196078 14.62745098 14.85294118 15.07843137 15.30392157 15.52941176\n",
+      " 15.75490196 15.98039216 16.20588235 16.43137255 16.65686275 16.88235294\n",
+      " 17.10784314 17.33333333 17.55882353 17.78431373 18.00980392 18.23529412\n",
+      " 18.46078431 18.68627451 18.91176471 19.1372549  19.3627451  19.58823529\n",
+      " 19.81372549 20.03921569 20.26470588 20.49019608 20.71568627 20.94117647\n",
+      " 21.16666667 21.39215686 21.61764706 21.84313725 22.06862745 22.29411765\n",
+      " 22.51960784 22.74509804 22.97058824 23.19607843 23.42156863 23.64705882\n",
+      " 23.87254902 24.09803922 24.32352941 24.54901961 24.7745098 ]\n"
+     ]
+    }
+   ],
+   "execution_count": 4
   }
  ],
  "metadata": {