sympy-Assignment4-Pb7.ipynb

sympy-assignment4-pb7.ipynb

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            "\u001b[0;36m  File \u001b[0;32m\"<ipython-input-1-299510c2fa03>\"\u001b[0;36m, line \u001b[0;32m1\u001b[0m\n\u001b[0;31m    <h1>\u001b[0m\n\u001b[0m    ^\u001b[0m\n\u001b[0;31mSyntaxError\u001b[0m\u001b[0;31m:\u001b[0m invalid syntax\n"
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      "cell_type": "markdown",
      "source": [
        "<h1> Problem 7 from Assignment 4"
      ],
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      "source": [
        "# Attempt 2 using help from Sympy documentation\n",
        "from sympy import Function, dsolve, Derivative, checkodesol\n",
        "from sympy.abc import t\n",
        "y = Function('y')\n",
        "result = dsolve(Derivative(y(t), t, t) + 2*Derivative(y(t),t)+26*y(t), y(t))\n",
        "checkodesol(Derivative(y(t), t, t) + 2*Derivative(y(t),t)+26*y(t), result)\n"
      ],
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              "(True, 0)"
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          "metadata": {},
          "execution_count": 23
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      "execution_count": 24,
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          "data": {
            "text/plain": [
              "Eq(y(t), (C1*sin(5*t) + C2*cos(5*t))*exp(-t))"
            ],
            "text/latex": "$\\displaystyle y{\\left(t \\right)} = \\left(C_{1} \\sin{\\left(5 t \\right)} + C_{2} \\cos{\\left(5 t \\right)}\\right) e^{- t}$"
          },
          "metadata": {},
          "execution_count": 24
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "constants = result.subs(t, 0).subs(y(0), 6).subs(Derivative(y(t)).subs(t, 0), 0)"
      ],
      "metadata": {
        "id": "6Zn6CfKF0rBI"
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      "execution_count": 27,
      "outputs": []
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      "source": [
        "constants"
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      "execution_count": 28,
      "outputs": [
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          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "Eq(6, C2)"
            ],
            "text/latex": "$\\displaystyle 6 = C_{2}$"
          },
          "metadata": {},
          "execution_count": 28
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# Attempt 2 using help from Sympy documentation\n",
        "from sympy import Function, dsolve, Derivative, checkodesol\n",
        "from sympy.abc import t  # Independent variable\n",
        "x = Function('x')\n",
        "result = dsolve(Derivative(x(t), t, t) + 2*Derivative(x(t),t)+26*x(t), x(t),ics={x(t).diff(t).subs(t, 0): 0,x(t).subs(t, 0): 6})\n",
        "\n",
        "checkodesol(Derivative(x(t), t, t) + 2*Derivative(x(t),t)+26*x(t), result)"
      ],
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      },
      "execution_count": 31,
      "outputs": [
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          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "(True, 0)"
            ]
          },
          "metadata": {},
          "execution_count": 31
        }
      ]
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      "execution_count": 32,
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        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "Eq(x(t), (6*sin(5*t)/5 + 6*cos(5*t))*exp(-t))"
            ],
            "text/latex": "$\\displaystyle x{\\left(t \\right)} = \\left(\\frac{6 \\sin{\\left(5 t \\right)}}{5} + 6 \\cos{\\left(5 t \\right)}\\right) e^{- t}$"
          },
          "metadata": {},
          "execution_count": 32
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "<h1> Non-Homogeneous Equation x\" + 2x' + 26x -82 cos(4t) = 0"
      ],
      "metadata": {
        "id": "0VKXvK2-4DBJ"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "from sympy import Function, dsolve, Derivative, cos, checkodesol\n",
        "from sympy.abc import t  # Independent variable\n",
        "x = Function('x')\n",
        "result = dsolve(Derivative(x(t), t, t) + 2*Derivative(x(t),t)+26*x(t)-82*cos(4*t), x(t),ics={x(t).diff(t).subs(t, 0): 0,x(t).subs(t, 0): 6})\n",
        "checkodesol(Derivative(x(t), t, t) + 2*Derivative(x(t),t)+26*x(t)-82*cos(4*t), result)"
      ],
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        "id": "K4PMIITw4UkO",
        "outputId": "0fa90581-355b-4834-d404-da6ce6186036"
      },
      "execution_count": 33,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "(True, 0)"
            ]
          },
          "metadata": {},
          "execution_count": 33
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# The solution to the non-homogeneous equation is\n",
        "result"
      ],
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      },
      "execution_count": 34,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "Eq(x(t), (-3*sin(5*t) + cos(5*t))*exp(-t) + 4*sin(4*t) + 5*cos(4*t))"
            ],
            "text/latex": "$\\displaystyle x{\\left(t \\right)} = \\left(- 3 \\sin{\\left(5 t \\right)} + \\cos{\\left(5 t \\right)}\\right) e^{- t} + 4 \\sin{\\left(4 t \\right)} + 5 \\cos{\\left(4 t \\right)}$"
          },
          "metadata": {},
          "execution_count": 34
        }
      ]
    }
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}