Assignment4-5.ipynb

assignment4-5.ipynb

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        "<h4>Problem Statement - Assignment 4-5"
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      "source": [
        "Use the method of variation of parameters to solve the ODE: y\" + 9y = sin(3x)"
      ]
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      "source": [
        "<h4> Solve the Complementary Solution"
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            "text/plain": [
              "Eq(y(x), C1*sin(3*x) + C2*cos(3*x))"
            ],
            "text/latex": "$\\displaystyle y{\\left(x \\right)} = C_{1} \\sin{\\left(3 x \\right)} + C_{2} \\cos{\\left(3 x \\right)}$"
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          "metadata": {},
          "execution_count": 1
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      "source": [
        "# Import library\n",
        "from sympy import diff, dsolve, symbols, Function, sin, cos, integrate, simplify\n",
        "\n",
        "# Define the function and variables\n",
        "x = symbols('x')\n",
        "y = Function('y')(x)\n",
        "hode = y.diff(x,x) + 9*y # Homogeneous Equation for complementary function\n",
        "sol = dsolve(hode) #obtain the complementary function or the soln of the homogeneous ode\n",
        "sol"
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            "text/plain": [
              "-3"
            ],
            "text/latex": "$\\displaystyle -3$"
          },
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          "execution_count": 3
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      ],
      "source": [
        "# Calculate the Wronksian\n",
        "y1 = sin(3*x)\n",
        "y1p = y1.diff(x)\n",
        "y2 = cos(3*x)\n",
        "y2p = y2.diff(x)\n",
        "W = y1*y2p - y2*y1p\n",
        "simplify(W)"
      ]
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    {
      "cell_type": "markdown",
      "id": "22e4e064",
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      "source": [
        "Obtain the particular solution using the formula\n",
        "\n",
        "$ y_p = -y1(x) \\int \\frac{y2(x) f(x)}{W(x)}dx + y2(x) \\int \\frac{y1(x) f(x)}{W(x)}dx $"
      ]
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      "cell_type": "code",
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      "id": "7feca356",
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          "data": {
            "text/plain": [
              "-x*cos(3*x)/6"
            ],
            "text/latex": "$\\displaystyle - \\frac{x \\cos{\\left(3 x \\right)}}{6}$"
          },
          "metadata": {},
          "execution_count": 5
        }
      ],
      "source": [
        "Fx = sin(3*x)\n",
        "yp1 = -y1* integrate(y2*Fx/W) # This is the first part of the Particular solution\n",
        "yp2 = y2* integrate(y1*Fx/W) # this is the second part\n",
        "yp = yp1 + yp2\n",
        "yp\n",
        "simplify(yp)  # This is my particular solution"
      ]
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          "data": {
            "text/plain": [
              "Eq(y(x), C2*sin(3*x) + (C1 - x/6)*cos(3*x))"
            ],
            "text/latex": "$\\displaystyle y{\\left(x \\right)} = C_{2} \\sin{\\left(3 x \\right)} + \\left(C_{1} - \\frac{x}{6}\\right) \\cos{\\left(3 x \\right)}$"
          },
          "metadata": {},
          "execution_count": 11
        }
      ],
      "source": [
        "# Check using direct solution:\n",
        "odex = hode - Fx\n",
        "dsolve(odex)"
      ]
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      "source": [
        "Note the constant 1/18 for the sin(3x) in the particular solution is subsumed within C2.  "
      ]
    },
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