MAT120_LAB_Lec05.ipynb

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    "kernelspec": {
      "name": "python3",
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  "cells": [
    {
      "cell_type": "code",
      "execution_count": 2,
      "metadata": {
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        },
        "id": "YOEBvaYIvdUd",
        "outputId": "baeace64-02ec-4b01-9997-ffa6a35bff7f"
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      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "pi"
            ],
            "text/latex": "$\\displaystyle \\pi$"
          },
          "metadata": {},
          "execution_count": 2
        }
      ],
      "source": [
        "import sympy as sp\n",
        "sp.pi\n"
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "x,t=sp.symbols('x t')\n",
        "# Declare a function\n",
        "y=sp.Function('y')(x)\n",
        "y"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
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        },
        "id": "StO0-CDmv4Xr",
        "outputId": "8d5a66f3-aa7e-4b03-d547-6375131362dc"
      },
      "execution_count": 3,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "y(x)"
            ],
            "text/latex": "$\\displaystyle y{\\left(x \\right)}$"
          },
          "metadata": {},
          "execution_count": 3
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "Solve the following differential equation\n",
        "# $dy/dx = x^2 + 5x $"
      ],
      "metadata": {
        "id": "24rxWccBoUse"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# First write the differential equation\n",
        "ode1=sp.Eq(sp.diff(y,x,1),(x**2+5*x))\n",
        "ode1"
      ],
      "metadata": {
        "colab": {
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        },
        "id": "zHY1xLQEw0Kh",
        "outputId": "b39278d8-0c42-4d38-d179-d720d33e33be"
      },
      "execution_count": 5,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "Eq(Derivative(y(x), x), x**2 + 5*x)"
            ],
            "text/latex": "$\\displaystyle \\frac{d}{d x} y{\\left(x \\right)} = x^{2} + 5 x$"
          },
          "metadata": {},
          "execution_count": 5
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "sp.dsolve(ode1)"
      ],
      "metadata": {
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        },
        "id": "j0TmRMYQxo7_",
        "outputId": "09f1eb2d-cfcf-4b21-bcc7-28df512cb78b"
      },
      "execution_count": 6,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "Eq(y(x), C1 + x**3/3 + 5*x**2/2)"
            ],
            "text/latex": "$\\displaystyle y{\\left(x \\right)} = C_{1} + \\frac{x^{3}}{3} + \\frac{5 x^{2}}{2}$"
          },
          "metadata": {},
          "execution_count": 6
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "Solve the second order differential equation\n",
        "# $d^2y/dt^2+dy/dt-6 =0 $"
      ],
      "metadata": {
        "id": "egtDXdJ0pPyc"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "y=sp.Function('y')(t)\n",
        "ode2=sp.Eq((sp.diff(y,t,2)+sp.diff(y,t,1)-6),0)\n",
        "ode2"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 54
        },
        "id": "1OtBe0X2xrrf",
        "outputId": "82e1633a-01a2-438f-9eb5-d9f4bb33eb62"
      },
      "execution_count": 7,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "Eq(Derivative(y(t), t) + Derivative(y(t), (t, 2)) - 6, 0)"
            ],
            "text/latex": "$\\displaystyle \\frac{d}{d t} y{\\left(t \\right)} + \\frac{d^{2}}{d t^{2}} y{\\left(t \\right)} - 6 = 0$"
          },
          "metadata": {},
          "execution_count": 7
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "sp.dsolve(ode2).simplify()"
      ],
      "metadata": {
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          "height": 39
        },
        "id": "Y9G0gfyczX5O",
        "outputId": "20c4e98f-c479-4063-8259-0f7897e96589"
      },
      "execution_count": 8,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "Eq(y(t), C1 + C2*exp(-t) + 6*t)"
            ],
            "text/latex": "$\\displaystyle y{\\left(t \\right)} = C_{1} + C_{2} e^{- t} + 6 t$"
          },
          "metadata": {},
          "execution_count": 8
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "Differential equation with intial condition\n",
        "# $dy/dt=t^2 $\n",
        "# $y(0)=1 $"
      ],
      "metadata": {
        "id": "HQHwBbiesH9X"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "y=sp.Function('y')(t)\n",
        "ode3=sp.Eq(sp.diff(y,t,1),t**2)\n",
        "ode3"
      ],
      "metadata": {
        "id": "gfVu0eLqzb2u",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 52
        },
        "outputId": "70888141-955c-418c-b3a1-9035c898c64a"
      },
      "execution_count": 10,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "Eq(Derivative(y(t), t), t**2)"
            ],
            "text/latex": "$\\displaystyle \\frac{d}{d t} y{\\left(t \\right)} = t^{2}$"
          },
          "metadata": {},
          "execution_count": 10
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# Pass the initial condition as a dictionary\n",
        "sp.dsolve(ode3,ics={y.subs(t,0):'1'})"
      ],
      "metadata": {
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          "base_uri": "https://localhost:8080/",
          "height": 54
        },
        "id": "ArXaGC4n0--5",
        "outputId": "a9e92a28-ad13-4b2f-9281-63f278a146a6"
      },
      "execution_count": 12,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "Eq(y(t), t**3/3 + 1)"
            ],
            "text/latex": "$\\displaystyle y{\\left(t \\right)} = \\frac{t^{3}}{3} + 1$"
          },
          "metadata": {},
          "execution_count": 12
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "Solve the differential equation of simple harmonic motion with initial conditions\n",
        "# $dy^2/dt^2=-\\omega^2y $\n",
        "# $y(0)=0 $\n",
        "# $y'(0)=1 $"
      ],
      "metadata": {
        "id": "1fbeqzwhtV8D"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# Define the symbol omega and save it in a variable say w\n",
        "w=sp.symbols('omega')"
      ],
      "metadata": {
        "id": "cLO_h38y2Ykl"
      },
      "execution_count": 14,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "# Write the ode and check it\n",
        "SHM=sp.Eq(sp.diff(y,t,2),-w**2*y)\n",
        "display(SHM)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 54
        },
        "id": "Umxnv-vM2nYF",
        "outputId": "b9b85996-2af9-440d-df33-b4dfce14e9ec"
      },
      "execution_count": 16,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "Eq(Derivative(y(t), (t, 2)), -omega**2*y(t))"
            ],
            "text/latex": "$\\displaystyle \\frac{d^{2}}{d t^{2}} y{\\left(t \\right)} = - \\omega^{2} y{\\left(t \\right)}$"
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# Solution of the equation without initial conditions\n",
        "sp.dsolve(SHM).simplify()"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 39
        },
        "id": "xrKWuEuI4tbF",
        "outputId": "de329af7-17df-45b1-b6ee-9350fbfa3b4c"
      },
      "execution_count": 17,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "Eq(y(t), C1*exp(-I*omega*t) + C2*exp(I*omega*t))"
            ],
            "text/latex": "$\\displaystyle y{\\left(t \\right)} = C_{1} e^{- i \\omega t} + C_{2} e^{i \\omega t}$"
          },
          "metadata": {},
          "execution_count": 17
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# Pass the initial conditions in ics\n",
        "soln=sp.dsolve(SHM,ics={y.subs(t,0):'0',sp.diff(y,t,1).subs(t,0):'1'}).simplify()\n",
        "soln"
      ],
      "metadata": {
        "colab": {
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          "height": 53
        },
        "id": "W19WDtA12jXe",
        "outputId": "1690ced1-23a5-455c-e315-6ef8092a2dfe"
      },
      "execution_count": 24,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "Eq(y(t), sin(omega*t)/omega)"
            ],
            "text/latex": "$\\displaystyle y{\\left(t \\right)} = \\frac{\\sin{\\left(\\omega t \\right)}}{\\omega}$"
          },
          "metadata": {},
          "execution_count": 24
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# Get the right hand side of the solution\n",
        "display(soln.rhs)"
      ],
      "metadata": {
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          "base_uri": "https://localhost:8080/",
          "height": 53
        },
        "id": "272ZohrKx_Ek",
        "outputId": "28ee066a-7249-4af3-eada-e97b777ec392"
      },
      "execution_count": 43,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "sin(omega*t)/omega"
            ],
            "text/latex": "$\\displaystyle \\frac{\\sin{\\left(\\omega t \\right)}}{\\omega}$"
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "Say $\\omega=\\pi/6$. Display the equation of SHM and see how it's look like."
      ],
      "metadata": {
        "id": "e2BaKEemvo5O"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "from sympy import *\n",
        "value=sp.pi/6"
      ],
      "metadata": {
        "id": "6JZZBwdF7RKk"
      },
      "execution_count": 44,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "soln.subs(w,value)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 56
        },
        "id": "aA4VCNuu6yFU",
        "outputId": "9b3fb2d0-423c-4e68-94ff-bc32542b3cf2"
      },
      "execution_count": 45,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "Eq(y(t), 6*sin(pi*t/6)/pi)"
            ],
            "text/latex": "$\\displaystyle y{\\left(t \\right)} = \\frac{6 \\sin{\\left(\\frac{\\pi t}{6} \\right)}}{\\pi}$"
          },
          "metadata": {},
          "execution_count": 45
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "sp.plot(soln.subs(w,value).rhs)"
      ],
      "metadata": {
        "id": "0r7TG7Em2hGd",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 504
        },
        "outputId": "5570df37-2997-4e55-fac0-e1ef53674b4d"
      },
      "execution_count": 46,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 640x480 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {}
        },
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "<sympy.plotting.plot.Plot at 0x7ee66314b040>"
            ]
          },
          "metadata": {},
          "execution_count": 46
        }
      ]
    }
  ]
}