Assignment4-Problem2.ipynb

assignment4-problem2.ipynb

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      "source": [
        "# <h4> Assignment 4: Problem 2"
      ],
      "metadata": {
        "id": "L6a0JlACJFBY"
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    {
      "cell_type": "markdown",
      "source": [
        "Find the particular solution of 3y” + y’-2y = 2cos(x)."
      ],
      "metadata": {
        "id": "MoRea56dJOPB"
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      "cell_type": "markdown",
      "source": [
        "# <h5> Solution Strategy:"
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        "\n",
        "\n",
        "1.  Check if the characteristic equation less than zero.  If it is then there will be a cos term in the complementary solution.\n",
        "2.   If there is no cos(x) term in the complementary solution the guess for yp = Acos(x) + Bsin(x) - Make sure you include both sin and cosine\n",
        "3.  If there is a cos(x) term in the complementary solution then use the guess Axcos(x) + Bsin(x) as the guess and proceed.  \n",
        "\n"
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      "source": [
        "<hr>"
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    {
      "cell_type": "markdown",
      "source": [
        "# <h5> Code"
      ],
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    {
      "cell_type": "code",
      "source": [
        "# Import libraries\n",
        "import numpy as np\n",
        "import sympy as sp"
      ],
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        "id": "2DrQUCsCLN8p"
      },
      "execution_count": 35,
      "outputs": []
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      "cell_type": "code",
      "source": [
        "# Calculate the value of the discriminant function\n",
        "a = 3\n",
        "b = 1\n",
        "c = -2\n",
        "disc = np.sqrt(np.power(b,2)-4*a*c)\n",
        "if(disc<0):\n",
        "  print(\"Cosine term in Complementary Solution: Guess = (AxCos(x) + BxSin(x))\")\n",
        "else:\n",
        "  print(\"No cosine term in Complementary solution: Guess = (ACos(x)+BSin(x))\")"
      ],
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          "text": [
            "No cosine term in Complementary solution: Guess = (ACos(x)+BSin(x))\n"
          ]
        }
      ]
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      "cell_type": "code",
      "source": [
        "# Assume the Particular solution is Acos(x) + B sin(x)\n",
        "x= sp.symbols('x')\n",
        "A = sp.symbols('A')\n",
        "B = sp.symbols('B')\n",
        "y = sp.Function('y')\n",
        "y = A*sp.cos(x)+B*sp.sin(x)  # define function\n",
        "y1 = y.diff(x) # Get the first derivative\n",
        "y2 = y.diff(x,x) # Get the second derivative term\n",
        "\n"
      ],
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        "id": "nsOrnoNBMr6S"
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      "execution_count": 44,
      "outputs": []
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      "cell_type": "code",
      "source": [
        "ode = 3*y2 + y1 - 2*y-2*sp.cos(x) # Write the ODE bring the Forcing function from RHS to LHS\n",
        "sp.solve(odex,[A,B]) # Solve non-homogeneous ODE for unknowns A and B of the particular solution"
      ],
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      "outputs": [
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          "data": {
            "text/plain": [
              "{A: -5/13, B: 1/13}"
            ]
          },
          "metadata": {},
          "execution_count": 47
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