Added(A)/Deleted(D) following books

 A  Basic_mechanical_engineering_by_Basant_Agrawal_,_C.M_Agrawal/README.txt
A  Numerical_Methods_by_E._Balaguruswamy/chapter10.ipynb
A  Numerical_Methods_by_E._Balaguruswamy/chapter11.ipynb
A  Numerical_Methods_by_E._Balaguruswamy/chapter12.ipynb
A  Numerical_Methods_by_E._Balaguruswamy/chapter13.ipynb
A  Numerical_Methods_by_E._Balaguruswamy/chapter14.ipynb
A  Numerical_Methods_by_E._Balaguruswamy/chapter15.ipynb
A  Numerical_Methods_by_E._Balaguruswamy/chapter3.ipynb
A  Numerical_Methods_by_E._Balaguruswamy/chapter4.ipynb
A  Numerical_Methods_by_E._Balaguruswamy/chapter6.ipynb
A  Numerical_Methods_by_E._Balaguruswamy/chapter7.ipynb
A  Numerical_Methods_by_E._Balaguruswamy/chapter8.ipynb
A  Numerical_Methods_by_E._Balaguruswamy/chapter9.ipynb
A  Numerical_Methods_by_E._Balaguruswamy/screenshots/greatest-precision-4.png
A  Numerical_Methods_by_E._Balaguruswamy/screenshots/rounding-off-4.png
A  Numerical_Methods_by_E._Balaguruswamy/screenshots/truncation-error-4.png
A  Physics_BSc(Paper_4)_by_Sanjeeva_Rao,_Bhikshmaiah,_Ramakrishna_Reddy,_Ananta_Ramaiah/C_4JhcI7F.ipynb
A  Physics_BSc(Paper_4)_by_Sanjeeva_Rao,_Bhikshmaiah,_Ramakrishna_Reddy,_Ananta_Ramaiah/C_5CfOfQx.ipynb
A  Physics_BSc(Paper_4)_by_Sanjeeva_Rao,_Bhikshmaiah,_Ramakrishna_Reddy,_Ananta_Ramaiah/C_B30VPml.ipynb
A  Physics_BSc(Paper_4)_by_Sanjeeva_Rao,_Bhikshmaiah,_Ramakrishna_Reddy,_Ananta_Ramaiah/C_BE8QjoS.ipynb
A  Physics_BSc(Paper_4)_by_Sanjeeva_Rao,_Bhikshmaiah,_Ramakrishna_Reddy,_Ananta_Ramaiah/C_Gb015bo.ipynb
A  Physics_BSc(Paper_4)_by_Sanjeeva_Rao,_Bhikshmaiah,_Ramakrishna_Reddy,_Ananta_Ramaiah/C_KQ8ycMr.ipynb
A  Physics_BSc(Paper_4)_by_Sanjeeva_Rao,_Bhikshmaiah,_Ramakrishna_Reddy,_Ananta_Ramaiah/C_LIZWeY4.ipynb
A  Physics_BSc(Paper_4)_by_Sanjeeva_Rao,_Bhikshmaiah,_Ramakrishna_Reddy,_Ananta_Ramaiah/C_Opo6g3L.ipynb
A  Physics_BSc(Paper_4)_by_Sanjeeva_Rao,_Bhikshmaiah,_Ramakrishna_Reddy,_Ananta_Ramaiah/C_Pq40WOu.ipynb
A  Physics_BSc(Paper_4)_by_Sanjeeva_Rao,_Bhikshmaiah,_Ramakrishna_Reddy,_Ananta_Ramaiah/C_QLXi0uM.ipynb
A  Physics_BSc(Paper_4)_by_Sanjeeva_Rao,_Bhikshmaiah,_Ramakrishna_Reddy,_Ananta_Ramaiah/C_TdOIeIQ.ipynb
A  Physics_BSc(Paper_4)_by_Sanjeeva_Rao,_Bhikshmaiah,_Ramakrishna_Reddy,_Ananta_Ramaiah/C_UiM06tF.ipynb
A  Physics_BSc(Paper_4)_by_Sanjeeva_Rao,_Bhikshmaiah,_Ramakrishna_Reddy,_Ananta_Ramaiah/C_VQabJzR.ipynb
A  Physics_BSc(Paper_4)_by_Sanjeeva_Rao,_Bhikshmaiah,_Ramakrishna_Reddy,_Ananta_Ramaiah/C_XKPPUxj.ipynb
A  Physics_BSc(Paper_4)_by_Sanjeeva_Rao,_Bhikshmaiah,_Ramakrishna_Reddy,_Ananta_Ramaiah/C_dfFlLnm.ipynb
A  Physics_BSc(Paper_4)_by_Sanjeeva_Rao,_Bhikshmaiah,_Ramakrishna_Reddy,_Ananta_Ramaiah/C_fegIkl6.ipynb
A  Physics_BSc(Paper_4)_by_Sanjeeva_Rao,_Bhikshmaiah,_Ramakrishna_Reddy,_Ananta_Ramaiah/C_g1GxlUN.ipynb
A  Physics_BSc(Paper_4)_by_Sanjeeva_Rao,_Bhikshmaiah,_Ramakrishna_Reddy,_Ananta_Ramaiah/C_gGvuusH.ipynb
A  Physics_BSc(Paper_4)_by_Sanjeeva_Rao,_Bhikshmaiah,_Ramakrishna_Reddy,_Ananta_Ramaiah/C_gZeNd4b.ipynb
A  Physics_BSc(Paper_4)_by_Sanjeeva_Rao,_Bhikshmaiah,_Ramakrishna_Reddy,_Ananta_Ramaiah/C_khAt4Y6.ipynb
A  Physics_BSc(Paper_4)_by_Sanjeeva_Rao,_Bhikshmaiah,_Ramakrishna_Reddy,_Ananta_Ramaiah/C_oJQM5Mb.ipynb
A  Physics_BSc(Paper_4)_by_Sanjeeva_Rao,_Bhikshmaiah,_Ramakrishna_Reddy,_Ananta_Ramaiah/C_p22tFeA.ipynb
A  Physics_BSc(Paper_4)_by_Sanjeeva_Rao,_Bhikshmaiah,_Ramakrishna_Reddy,_Ananta_Ramaiah/C_qNHcrb8.ipynb
A  Physics_BSc(Paper_4)_by_Sanjeeva_Rao,_Bhikshmaiah,_Ramakrishna_Reddy,_Ananta_Ramaiah/C_qeUxd0G.ipynb
A  Physics_BSc(Paper_4)_by_Sanjeeva_Rao,_Bhikshmaiah,_Ramakrishna_Reddy,_Ananta_Ramaiah/C_r0nfWNs.ipynb
A  Physics_BSc(Paper_4)_by_Sanjeeva_Rao,_Bhikshmaiah,_Ramakrishna_Reddy,_Ananta_Ramaiah/C_uWQUwaW.ipynb
A  Physics_BSc(Paper_4)_by_Sanjeeva_Rao,_Bhikshmaiah,_Ramakrishna_Reddy,_Ananta_Ramaiah/C_vkMAnbH.ipynb
A  Physics_BSc(Paper_4)_by_Sanjeeva_Rao,_Bhikshmaiah,_Ramakrishna_Reddy,_Ananta_Ramaiah/C_wP2wGMS.ipynb
A  Physics_BSc(Paper_4)_by_Sanjeeva_Rao,_Bhikshmaiah,_Ramakrishna_Reddy,_Ananta_Ramaiah/C_x09tDSO.ipynb
A  Physics_BSc(Paper_4)_by_Sanjeeva_Rao,_Bhikshmaiah,_Ramakrishna_Reddy,_Ananta_Ramaiah/C_yCW2orc.ipynb

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diff --git a/Numerical_Methods_by_E._Balaguruswamy/chapter13.ipynb b/Numerical_Methods_by_E._Balaguruswamy/chapter13.ipynb
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+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Chapter 13 - Numerical solution of ordinary differential equations"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Example No. 13_01 Pg No. 414"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      " y(0.25) =  5.3125\n",
+      " y(0.5) =  7.40685424949\n"
+     ]
+    }
+   ],
+   "source": [
+    "from __future__ import division\n",
+    "from math import sqrt\n",
+    "#Taylor method\n",
+    "\n",
+    "def f(x,y):\n",
+    "    F = x**2 + y**2\n",
+    "    return F\n",
+    "def d2y(x,y):\n",
+    "    D2Y = 2*x + 2*y*f(x,y)\n",
+    "    return D2Y\n",
+    "def d3y(x,y):\n",
+    "    D3Y = 2 + 2*y*d2y(x,y) + 2*f(x,y)**2\n",
+    "    return D3Y\n",
+    "def y(x):\n",
+    "    Y = 1 + f(0,1)*x + d2y(0,1)*x**2/2 + d3y(0,1)*sqrt(x)\n",
+    "    return Y\n",
+    "print ' y(0.25) = ',y(0.25)\n",
+    "print ' y(0.5) = ',y(0.5)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Example No. 13_02 Pg No. 415"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      " Iteration-1\n",
+      "\n",
+      " dy(0) = 0.000000\n",
+      " d2y(0) = 0.000000\n",
+      " d3y(0) = 2.000000\n",
+      " d4y(0) = 0.000000\n",
+      " \n",
+      "y(0) = 0.002667\n",
+      "\n",
+      "\n",
+      " Iteration-2\n",
+      "\n",
+      " dy(0) = 0.040007\n",
+      " d2y(0) = 0.400213\n",
+      " d3y(0) = 2.005336\n",
+      " d4y(0) = 0.106763\n",
+      " \n",
+      "y(0) = 0.021353\n",
+      "\n",
+      "\n"
+     ]
+    }
+   ],
+   "source": [
+    "from __future__ import division\n",
+    "from scipy.misc import factorial\n",
+    "#Recursive Taylor Method\n",
+    "\n",
+    "def f(x,y):\n",
+    "    F = x**2 + y**2\n",
+    "    return F\n",
+    "def d2y(x,y):\n",
+    "    D2Y = 2*x + 2*y*f(x,y)\n",
+    "    return D2Y\n",
+    "def d3y(x,y):\n",
+    "    D3Y = 2 + 2*y*d2y(x,y) + 2*f(x,y)**2\n",
+    "    return D3Y\n",
+    "def d4y(x,y):\n",
+    "    D4Y = 6*f(x,y)*d2y(x,y) + 2*y*d3y(x,y)\n",
+    "    return D4Y\n",
+    "h = 0.2 #\n",
+    "def y(x,y):\n",
+    "    Y = y + f(x,y)*h + d2y(x,y)*h**2/2 + d3y(x,y)*h**3/6 + d4y(x,y)*h**4/factorial(4)\n",
+    "    return Y\n",
+    "x0 = 0#\n",
+    "y0 = 0 #\n",
+    "y_=[]\n",
+    "for i in range(1,3):\n",
+    "    y_.append(y(x0,y0))\n",
+    "    print ' Iteration-%d\\n\\n dy(0) = %f\\n d2y(0) = %f\\n d3y(0) = %f\\n d4y(0) = %f\\n '%(i,f(x0,y0),d2y(x0,y0),d3y(x0,y0),d4y(x0,y0)) \n",
+    "    x0 = x0 + i*h\n",
+    "    y0 = y_[i-1]\n",
+    "    print 'y(0) = %f\\n\\n'%(y_[i-1])"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Example No. 13_03 Pg No. 417"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "for dy(x) = x**2 + y**2 the results are :\n",
+      "y(0.1) =  0.000333334920635\n",
+      "y(0.2) =  0.00266686984127\n",
+      "y(1) =  0.349206349206\n",
+      "for dy(x) = x*e**y the results are \n",
+      "y(0.1) =  0.0050125208594\n",
+      "y(0.2) =  0.0202013400268\n",
+      "y(1) =  0.6487212707\n"
+     ]
+    }
+   ],
+   "source": [
+    "from __future__ import division\n",
+    "from math import exp \n",
+    "#Picard's Method\n",
+    "\n",
+    "#y'(x) = x**2 + y**2,y(0) = 0\n",
+    "#y(1) = y0 + integral(x**2 + y0**2,x0,x)\n",
+    "#y(1) = x**3/3\n",
+    "#y(2) = 0 + integral(xY2 + y1**2,x0,x)\n",
+    "#     = integral(x**2 + x**6/9,0,x) = x**3/3 + x**7/63\n",
+    "#therefore y(x) = x**3 /3 + x**7/63\n",
+    "def y(x):\n",
+    "    Y = x**3/3 + x**7/63 \n",
+    "    return Y\n",
+    "print 'for dy(x) = x**2 + y**2 the results are :'\n",
+    "print 'y(0.1) = ',y(0.1)\n",
+    "print 'y(0.2) = ',y(0.2)\n",
+    "print 'y(1) = ',y(1)\n",
+    "\n",
+    "#y'(x) = x*e**y, y(0) = 0\n",
+    "#y0 = 0 , x0 = 0\n",
+    "#Y(1) = 0 + integral(x*e**0,0,x) = x**2/2\n",
+    "#y(2) = 0 + integral( x*e**( x**2/2 ) ,0,x) = e**(x**2/2)-1\n",
+    "#therefore y(x) = e**(x**2/2) - 1\n",
+    "def y(x):\n",
+    "    Y = exp(x**2/2) - 1 \n",
+    "    return Y\n",
+    "print 'for dy(x) = x*e**y the results are '                 \n",
+    "print 'y(0.1) = ',y(0.1)\n",
+    "print 'y(0.2) = ',y(0.2)\n",
+    "print 'y(1) = ',y(1)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Example No. 13_04 Pg No. 417"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "for h = 0.500000\n",
+      "\n",
+      "y(1.500000) = 4.000000\n",
+      "\n",
+      "y(2.000000) = 7.875000\n",
+      "\n",
+      "\n",
+      "for h = 0.250000\n",
+      "\n",
+      "y(1.250000) = 3.000000\n",
+      "\n",
+      "y(1.500000) = 4.421875\n",
+      "\n",
+      "y(1.750000) = 6.359375\n",
+      "\n",
+      "y(2.000000) = 8.906250\n",
+      "\n"
+     ]
+    }
+   ],
+   "source": [
+    "from __future__ import division\n",
+    "\n",
+    "#Euler's Method\n",
+    "\n",
+    "def dy(x):\n",
+    "    DY = 3*x**2 + 1\n",
+    "    return DY\n",
+    "x0 = 1\n",
+    "y = [0,2] #\n",
+    "#h = 0.5\n",
+    "h = 0.5\n",
+    "print 'for h = %f\\n'%(h)\n",
+    "for i in range(2,4):\n",
+    "    y.append(y[i-1] + h*dy(x0+(i-2)*h))\n",
+    "    print 'y(%f) = %f\\n'%(x0+(i-1)*h,y[i])\n",
+    "\n",
+    "#h = 0.25\n",
+    "h = 0.25\n",
+    "print '\\nfor h = %f\\n'%(h)\n",
+    "y = [0,2] #\n",
+    "for i in range(2,6):\n",
+    "    y.append(y[i-1] + h*dy(x0+(i-2)*h))\n",
+    "    print 'y(%f) = %f\\n'%(x0+(i-1)*h,y[i])"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Example No. 13_05 Pg No. 422"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "\n",
+      " Step 1 \n",
+      " x(1) = 1.500000\n",
+      " y(1.500000) = 4.000000\n",
+      "\n",
+      "\n",
+      " Et(1) = 0.875000\n",
+      "\n",
+      "\n",
+      " Step 2 \n",
+      " x(2) = 2.000000\n",
+      " y(2.000000) = 7.875000\n",
+      "\n",
+      "\n",
+      " Et(2) = 1.250000\n",
+      "\n",
+      "x      Est y   true y    Et      Globalerr\n",
+      "1.5 \t4.0 \t4.875 \t0.875 \t0.875\n",
+      "2.0 \t7.875 \t10.0 \t1.25 \t2.125\n"
+     ]
+    }
+   ],
+   "source": [
+    "from numpy import nditer\n",
+    "from __future__ import division\n",
+    "#Error estimation in Euler's Method\n",
+    "\n",
+    "def dy(x):\n",
+    "    DY = 3*x**2 + 1\n",
+    "    return DY\n",
+    "def d2y(x):\n",
+    "    D2Y = 6*x\n",
+    "    return D2Y\n",
+    "def d3y(x):\n",
+    "    D3Y = 6\n",
+    "    return D3Y\n",
+    "def exacty(x):\n",
+    "    Exacty = x**3 + x\n",
+    "    return Exacty\n",
+    "x0 = 1\n",
+    "y = 2\n",
+    "h = 0.5\n",
+    "X=[];ESTY=[];TRUEY=[];ET=[];GERR=[];\n",
+    "for i in range(2,4):\n",
+    "    x = x0 + (i-1)*h\n",
+    "    y = y + h*dy(x0+(i-2)*h)\n",
+    "    print '\\n Step %d \\n x(%d) = %f\\n y(%f) = %f\\n'%(i-1,i-1,x,x,y)\n",
+    "    Et = d2y(x0+(i-2)*h)*h**2/2 +  d3y(x0+(i-2)*h)*h**3/6\n",
+    "    print '\\n Et(%d) = %f\\n'%(i-1,Et)\n",
+    "    truey = exacty(x0+(i-1)*h)\n",
+    "    gerr = truey - y\n",
+    "    X.append(x)\n",
+    "    ESTY.append(y)\n",
+    "    TRUEY.append(truey)\n",
+    "    ET.append(Et)\n",
+    "    GERR.append(gerr)\n",
+    "\n",
+    "#table = [x y(2:3) truey Et gerr]\n",
+    "print 'x      Est y   true y    Et      Globalerr'    \n",
+    "for a,b,c,d,e in nditer([X,ESTY,TRUEY,ET,GERR]):\n",
+    "    print a,'\\t',b,'\\t',c,'\\t',d,'\\t',e"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Example No. 13_06 Pg No. 427"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "EULERS METHOD\n",
+      "y(1.250000) = 3.000000 \n",
+      " \n",
+      "y(1.500000) = 4.200000 \n",
+      " \n",
+      "y(1.750000) = 5.600000 \n",
+      " \n",
+      "y(2.000000) = 7.200000 \n",
+      " \n",
+      "HEUNS METHOD\n",
+      "\n",
+      "Iteration 1 \n",
+      " m1 = 4.000000\n",
+      " ye(1.250000) = 3.000000 \n",
+      " m2 = 4.800000 \n",
+      " y(1.250000) = 3.100000 \n",
+      "\n",
+      "\n",
+      "Iteration 2 \n",
+      " m1 = 4.960000\n",
+      " ye(1.500000) = 4.340000 \n",
+      " m2 = 5.786667 \n",
+      " y(1.500000) = 4.443333 \n",
+      "\n",
+      "\n",
+      "Iteration 3 \n",
+      " m1 = 5.924444\n",
+      " ye(1.750000) = 5.924444 \n",
+      " m2 = 6.770794 \n",
+      " y(1.750000) = 6.030238 \n",
+      "\n",
+      "\n",
+      "Iteration 4 \n",
+      " m1 = 6.891701\n",
+      " ye(2.000000) = 7.753163 \n",
+      " m2 = 7.753163 \n",
+      " y(2.000000) = 7.860846 \n",
+      "\n",
+      "x     Eulers   Heuns   \t\tAnalytical\n",
+      "0.0 \t0.0 \t0.000000 \t0.0\n",
+      "1.0 \t2.0 \t2.000000 \t2.0\n",
+      "1.25 \t3.0 \t3.100000 \t2.2360679775\n",
+      "1.5 \t4.2 \t4.443333 \t2.44948974278\n",
+      "1.75 \t5.6 \t6.030238 \t2.64575131106\n",
+      "2.0 \t7.2 \t7.860846 \t2.82842712475\n"
+     ]
+    }
+   ],
+   "source": [
+    "from numpy import nditer,sqrt\n",
+    "from __future__ import division\n",
+    "\n",
+    "#Heun's Method\n",
+    "\n",
+    "def f(x,y):\n",
+    "    F = 2*y/x \n",
+    "    return F\n",
+    "def exacty(x):\n",
+    "    Exacty = 2*sqrt(x)\n",
+    "    return Exacty\n",
+    "x = [0,1] #\n",
+    "y = [0,2] #\n",
+    "h = 0.25 #\n",
+    "#Euler's Method\n",
+    "print 'EULERS METHOD'\n",
+    "for i in range(2,6):\n",
+    "    x.append(x[i-1] + h )\n",
+    "    y.append(y[i-1] + h*f(x[i-1],y[i-1]))\n",
+    "    print 'y(%f) = %f \\n '%(x[i],y[i])\n",
+    "\n",
+    "eulery = y\n",
+    "#Heun's Method\n",
+    "print 'HEUNS METHOD'\n",
+    "ye=[0,0]\n",
+    "y = [0,2] #\n",
+    "for i in range(2,6):\n",
+    "    m1 = f(x[i-1],y[i-1]) #\n",
+    "    ye.append(y[i-1] + h*f(x[(i-1)],y[(i-1)]))\n",
+    "    m2 = f(x[(i)],ye[(i)]) \n",
+    "    y.append(y[(i-1)] + h*(m1 + m2)/2)\n",
+    "    print '\\nIteration %d \\n m1 = %f\\n ye(%f) = %f \\n m2 = %f \\n y(%f) = %f \\n'%(i-1,m1,x[(i)],ye[(i)],m2,x[(i)],y[(i)]) \n",
+    "    \n",
+    "    \n",
+    "truey = exacty(x) \n",
+    "print 'x     Eulers   Heuns   \\t\\tAnalytical'    \n",
+    "for a,b,c,d in nditer([x,eulery,y,truey]):\n",
+    "    print a,'\\t',b,'\\t%.6f'%c,'\\t',d\n",
+    "    "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Example No. 13_07 Pg NO. 433"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "y(1.250000) = 3.111111 \n",
+      " \n",
+      "y(1.500000) = 4.468687 \n",
+      " \n"
+     ]
+    }
+   ],
+   "source": [
+    "from __future__ import division\n",
+    "\n",
+    "#Polygon Method\n",
+    "\n",
+    "def f(x,y):\n",
+    "    F = 2*y/x\n",
+    "    return F\n",
+    "x=[1]\n",
+    "y=[2]\n",
+    "h = 0.25 #\n",
+    "for i in range(1,3):\n",
+    "    x.append(x[(i-1)] + h )\n",
+    "    y.append(y[(i-1)] + h*f(  x[(i-1)]+ h/2  ,  y[(i-1)] + h*f( x[(i-1)]  ,  y[(i-1)] )/2  ))\n",
+    "    print 'y(%f) = %f \\n '%(x[(i)],y[(i)])\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Example No. 13_08 Pg No. 439"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "\n",
+      "Iteration - 1\n",
+      " m1 = 0.000000\n",
+      " m2 = 0.010000 \n",
+      " m3 = 0.010001 \n",
+      " m4 = 0.040004 \n",
+      " y(0.200000) = 0.002667 \n",
+      "\n",
+      "\n",
+      "Iteration - 2\n",
+      " m1 = 0.040007\n",
+      " m2 = 0.090044 \n",
+      " m3 = 0.090136 \n",
+      " m4 = 0.160428 \n",
+      " y(0.400000) = 0.021360 \n",
+      "\n"
+     ]
+    }
+   ],
+   "source": [
+    "from __future__ import division\n",
+    "\n",
+    "#Classical Runge Kutta Method\n",
+    "\n",
+    "def f(x,y):\n",
+    "    F = x**2 + y**2\n",
+    "    return F\n",
+    "h = 0.2\n",
+    "x=[0]\n",
+    "y=[0]\n",
+    "\n",
+    "for i in range(0,2):\n",
+    "    m1 = f(  x[(i)]  ,  y[(i)]  ) #\n",
+    "    m2 = f(  x[i] + h/2  ,  y[(i)] + m1*h/2  ) #\n",
+    "    m3 = f(  x[(i)] + h/2  ,  y[(i)] + m2*h/2  ) #\n",
+    "    m4 = f(  x[(i)] + h  ,  y[(i)] + m3*h  ) #\n",
+    "    x.append(x[(i)] + h )\n",
+    "    y.append(y[(i)] +  (m1 + 2*m2 + 2*m3 + m4)*h/6 )\n",
+    "    \n",
+    "    print '\\nIteration - %d\\n m1 = %f\\n m2 = %f \\n m3 = %f \\n m4 = %f \\n y(%f) = %f \\n'%(i+1,m1,m2,m3,m4,x[(i+1)],y[(i+1)])\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Example No. 13_09 Pg No. 444"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "for four decimal places\n",
+      "h =  0.0143668085653\n",
+      "for six decimal places\n",
+      "h =  0.00454318377739\n",
+      "Note-We can use h = 0.01 for four decimal places and h = 0.004 for six decimal places\n"
+     ]
+    }
+   ],
+   "source": [
+    "from __future__ import division\n",
+    "\n",
+    "#Optimum step size\n",
+    "\n",
+    "x = 0.8 #\n",
+    "h1 = 0.05 #\n",
+    "y1 = 5.8410870 #\n",
+    "h2 = 0.025 #\n",
+    "y2 = 5.8479637 #\n",
+    "\n",
+    "#d = 4\n",
+    "h = ((h1**4 - h2**4)*10**(-4)/(2*(y2 - y1)))**(1/4)\n",
+    "print 'for four decimal places'\n",
+    "print 'h = ',h \n",
+    "\n",
+    "#d = 6\n",
+    "h = ((h1**4 - h2**4)*10**(-6)/(2*(y2 - y1)))**(1/4)\n",
+    "print 'for six decimal places'\n",
+    "print 'h = ' ,h\n",
+    "print 'Note-We can use h = 0.01 for four decimal places and h = 0.004 for six decimal places'"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Example No. 13_10 Pg NO. 446"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "yp4 =  8.00914285714\n",
+      "fp4 =  8.00914285714 yc4 =  8.00266666667\n",
+      "f4 =  8.00266666667\n",
+      "yc4 =  8.00212698413\n",
+      "Note- the exact solution is y(2) = 8\n"
+     ]
+    }
+   ],
+   "source": [
+    "from __future__ import division\n",
+    "\n",
+    "#Milne-Simpson Predictor-Corrector method\n",
+    "\n",
+    "def f(x,y):\n",
+    "    F = 2*y/x\n",
+    "    return F\n",
+    "x0 = 1 #\n",
+    "y0 = 2 #\n",
+    "h = 0.25 #\n",
+    "#Assuming y1 ,y2 and y3(required for milne-simpson formula) are estimated using Fourth- order Runge kutta method\n",
+    "x1 = x0 + h \n",
+    "y1 = 3.13 #\n",
+    "x2 = x1 + h \n",
+    "y2 = 4.5 #\n",
+    "x3 = x2 + h\n",
+    "y3 = 6.13 #\n",
+    "#Milne Predictor formula\n",
+    "yp4 = y0 + 4*h*(2*f(x1,y1) - f(x2,y2) + 2*f(x3,y3))/3\n",
+    "x4 = x3 + h \n",
+    "fp4 = f(x4,yp4) #\n",
+    "print 'yp4 = ',yp4\n",
+    "print 'fp4 = ',fp4, \n",
+    "#Simpson Corrector formula\n",
+    "yc4 = y2 + h*( f(x2,y2) + 4*f(x3,y3) + fp4)/3 \n",
+    "f4 = f(x4,yc4)\n",
+    "print 'yc4 = ',yc4\n",
+    "print 'f4 = ',f4 \n",
+    "\n",
+    "yc4 = y2 + h*( f(x2,y2) + 4*f(x3,y3) + f4)/3 \n",
+    "print 'yc4 = ',yc4 \n",
+    "print 'Note- the exact solution is y(2) = 8'"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Example No. 13_11 Pg NO. 446"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Adams Predictor formula\n",
+      "yp4 =  8.01135714286\n",
+      "fp4 =  8.01135714286\n",
+      "\n",
+      "Adams Corrector formula\n",
+      "yc4 =  8.00727901786\n",
+      "f4 =  8.00727901786\n",
+      "\n",
+      "refined-yc4 =  8.00689669364\n"
+     ]
+    }
+   ],
+   "source": [
+    "from __future__ import division\n",
+    "\n",
+    "#Adams-Bashforth-Moulton Method\n",
+    "\n",
+    "def f(x,y):\n",
+    "    F = 2*y/x\n",
+    "    return F\n",
+    "x0 = 1 #\n",
+    "y0 = 2 #\n",
+    "h = 0.25 #\n",
+    "x1 = x0 + h \n",
+    "y1 = 3.13 #\n",
+    "x2 = x1 + h \n",
+    "y2 = 4.5 #\n",
+    "x3 = x2 + h\n",
+    "y3 = 6.13 #\n",
+    "#Adams Predictor formula\n",
+    "yp4 = y3 + h*(55*f(x3,y3) - 59*f(x2,y2) + 37*f(x1,y1) - 9*f(x0,y0))/24\n",
+    "x4 = x3 + h \n",
+    "fp4 = f(x4,yp4) \n",
+    "print 'Adams Predictor formula'\n",
+    "print 'yp4 = ',yp4 \n",
+    "print 'fp4 = ',fp4 \n",
+    "#Adams Corrector formula\n",
+    "yc4 = y3 + h*( f(x1,y1) - 5*f(x2,y2) + 19*f(x3,y3) + 9*fp4 )/24 \n",
+    "f4 = f(x4,yc4)\n",
+    "print '\\nAdams Corrector formula'\n",
+    "print 'yc4 = ',yc4 \n",
+    "print 'f4 = ',f4 \n",
+    "\n",
+    "yc4 = y3 + h*( f(x1,y1) - 5*f(x2,y2) + 19*f(x3,y3) + 9*f4 )/24 \n",
+    "print '\\nrefined-yc4 = ',yc4"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Example No. 13_12 Pg No. 453"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "\n",
+      " y4p = 0.557351\n",
+      " f4p = -0.310640 \n",
+      " y4c = 0.555396 \n",
+      " Modifier Etc = 0.000067 \n",
+      " Modified y4c = 0.555464 \n",
+      "\n",
+      "\n",
+      " y5p = 0.500837\n",
+      " f5p = -0.250837 \n",
+      " y5c = 0.499959 \n",
+      " Modifier Etc = 0.000030 \n",
+      " Modified y5c = 0.499989 \n",
+      "\n",
+      "error =  1.11332736336e-05\n"
+     ]
+    }
+   ],
+   "source": [
+    "from __future__ import division\n",
+    "\n",
+    "#Milne-Simpson Method using modifier\n",
+    "\n",
+    "def f(y):\n",
+    "    F = -y**2\n",
+    "    return F\n",
+    "x = [ 1 , 1.2 , 1.4 , 1.6 ] \n",
+    "y = [ 1 , 0.8333333 , 0.7142857 , 0.625 ] \n",
+    "h = 0.2 #\n",
+    "\n",
+    "for i in range(0,2):\n",
+    "    yp = y[(i)] + 4*h*( 2*f( y[(i+1)] ) - f( y[(i+2)] ) + 2*f( y[(i+3)] ) )/3\n",
+    "    fp = f(yp) #\n",
+    "    yc = y[( i+2)] + h*(f( y[(i+2)] ) + 4*f( y[(i+3)] ) + fp )/3 \n",
+    "    Etc = -(yc - yp)/29\n",
+    "    y.append(yc + Etc)\n",
+    "    print '\\n y%dp = %f\\n f%dp = %f \\n y%dc = %f \\n Modifier Etc = %f \\n Modified y%dc = %f \\n'%(i+4,yp,i+4,fp,i+4,yc,Etc,i+4,y[(i+4)])\n",
+    "exactanswer = 0.5 #\n",
+    "err = exactanswer - y[5] #\n",
+    "print 'error = ',err"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Example No. 13_13 Pg No. 455"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "m1(1) = 0.000000\n",
+      " m1(2) = 1.000000\n",
+      " m2(1) = 0.200000\n",
+      " m2(2) = 1.100000\n",
+      " m(1) = 0.100000\n",
+      " m(2) = 1.050000\n",
+      " y1(0.1) = 1.010000\n",
+      " y2(0.1) = -0.895000\n",
+      "\n"
+     ]
+    }
+   ],
+   "source": [
+    "from __future__ import division\n",
+    "\n",
+    "#System of differetial Equations\n",
+    "\n",
+    "def f1(x,y1,y2):\n",
+    "    F1 = x + y1 + y2 \n",
+    "    return F1\n",
+    "def f2(x,y1,y2):\n",
+    "    F2 = 1 + y1 + y2 \n",
+    "    return F2\n",
+    "\n",
+    "x0 = 0 #\n",
+    "y10 = 1 #\n",
+    "y20 = -1 #\n",
+    "h = 0.1 #\n",
+    "m1= [f1( x0 , y10 , y20 )]\n",
+    "m1.append(f2( x0 , y10 , y20 ))\n",
+    "m2=[f1( x0+h , y10 + h*m1[0] , y20 + h*m1[1] )]\n",
+    "m2.append(f2( x0+h , y10 + h*m1[0] , y20 + h*m1[1] ))\n",
+    "m= [(m1[0] + m2[0])/2 ]\n",
+    "m.append((m1[1] + m2[1])/2)\n",
+    "\n",
+    "y1_0_1 = y10 + h*m[0]\n",
+    "y2_0_1 = y20 + h*m[1]\n",
+    "\n",
+    "print 'm1(1) = %f\\n m1(2) = %f\\n m2(1) = %f\\n m2(2) = %f\\n m(1) = %f\\n m(2) = %f\\n y1(0.1) = %f\\n y2(0.1) = %f\\n'%(m1[0],m1[1],m2[0],m2[1],m[0],m[1],y1_0_1,y2_0_1) "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Example No. 13_14 Pg No. 457"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "m1(1) = 1.000000\n",
+      " m1(2) = -2.000000\n",
+      " m2(1) = 0.600000\n",
+      " m2(2) = 0.600000\n",
+      " m(1) = 0.800000\n",
+      " m(2) = -0.700000\n",
+      " y1(0.1) = 0.160000\n",
+      " y2(0.1) = 0.860000\n",
+      "\n"
+     ]
+    }
+   ],
+   "source": [
+    "from __future__ import division\n",
+    "#Higher Order Differential Equations\n",
+    "\n",
+    "x0 = 0\n",
+    "y10 = 0\n",
+    "y20 = 1\n",
+    "h = 0.2\n",
+    "m1 = [y20] #\n",
+    "m1.append(6*x0 + 3*y10 - 2*y20)\n",
+    "m2= [y20 + h*m1[1]]\n",
+    "m2.append(6*(x0+h) + 3*(y10 + h*m1[0]) - 2*(y20 + h*m1[1]) )\n",
+    "m = [(m1[0] + m2[0])/2 ]\n",
+    "m.append((m1[1] + m2[1])/2)\n",
+    "\n",
+    "y1_0_2 = y10 + h*m[0]\n",
+    "y2_0_2 = y20 + h*m[1]\n",
+    "\n",
+    "print 'm1(1) = %f\\n m1(2) = %f\\n m2(1) = %f\\n m2(2) = %f\\n m(1) = %f\\n m(2) = %f\\n y1(0.1) = %f\\n y2(0.1) = %f\\n'%(m1[0],m1[1],m2[0],m2[1],m[0],m[1],y1_0_2,y2_0_2) "
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 2",
+   "language": "python",
+   "name": "python2"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 2
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython2",
+   "version": "2.7.9"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 0
+}