Assignment4-Problem6.ipynb

assignment4-problem6.ipynb

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        "<h4> Assignment 4 Problem 6 - Problem Statement"
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        "Verify by substitution that $ y_c = c_1x + c_2 \\frac{1}{x}$ is the complementary solution of the following differential equation $x^2y\" + xy' - y = 72x^5$\n",
        "\n",
        "**Complementary Solution Corresponds to the Homogeneous Part of the ODE**"
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        "# Import libraries\n",
        "import sympy as sp"
      ]
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            "text/latex": "$\\displaystyle 0$"
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        "# define variables and equation\n",
        "x = sp.symbols('x')  # define independent variable x\n",
        "c1 = sp.symbols('c1')  # define constant c1\n",
        "c2 = sp.symbols('c2') # define constant c2\n",
        "yc = c1*x + c2*1/x  # Write the complementary solution\n",
        "ypc = yc.diff(x)  # take the first derivative of yc\n",
        "yppc = yc.diff(x,x)  # take the second derivative of yc\n",
        "ode = x*x*yppc + x*ypc - yc # substite the yc, yc' and yc'' terms into the ode\n",
        "sp.simplify(ode) # simplify the expression to see if it results in a value of zero."
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        "As yc is the complementary solution, the substitution into the RHS of the ODE should lead to a value of zero.  "
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