Assignment4-8-9.ipynb

assignment4-8-9.ipynb

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      "source": [
        "<h4>Problem Statement - Assignment 4-8-9"
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        "Calculate the Wronksian (set up the equations) and solve for the particular solution of the ODE: y\" + 2y'+26y = 82cos(4t)"
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      "source": [
        "<h4> Solve the Complementary Solution"
      ]
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            "text/plain": [
              "Eq(x(t), (C1*sin(5*t) + C2*cos(5*t))*exp(-t))"
            ],
            "text/latex": "$\\displaystyle x{\\left(t \\right)} = \\left(C_{1} \\sin{\\left(5 t \\right)} + C_{2} \\cos{\\left(5 t \\right)}\\right) e^{- t}$"
          },
          "metadata": {},
          "execution_count": 2
        }
      ],
      "source": [
        "# Import library\n",
        "from sympy import diff, dsolve, symbols, Function, sin, cos, integrate, simplify,exp\n",
        "\n",
        "# Define the function and variables\n",
        "t = symbols('t') # Independent Variable\n",
        "x = Function('x')(t) # Dependent Variable\n",
        "hode = x.diff(t,t) + 2*x.diff(t)+ 26*x # Homogeneous Equation for complementary function\n",
        "sol = dsolve(hode) #obtain the complementary function\n",
        "sol"
      ]
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      "id": "b05bfcce",
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      "source": [
        "<h4> Equations for the Wronksian"
      ]
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      "source": [
        "From the complementary solution $y_1 = sin(5t)e^{-t}$ and $y_1 = cos(5t)e^{-t}$\n",
        "Let u and v be two functions of t which satisfy the following equations:\n",
        "\n",
        "$ u'y_1 + v'y_2 = 0 $\n",
        "\n",
        "$ u' y'_1 + v'y'_2 =82cos(4t) $\n"
      ]
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    {
      "cell_type": "code",
      "execution_count": 5,
      "id": "baf47fef",
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          "data": {
            "text/plain": [
              "-5*exp(-2*t)"
            ],
            "text/latex": "$\\displaystyle - 5 e^{- 2 t}$"
          },
          "metadata": {},
          "execution_count": 5
        }
      ],
      "source": [
        "# Calculate the Wronksian\n",
        "y1 = sin(5*t)*exp(-t)\n",
        "y1p = y1.diff(t)\n",
        "y2 = cos(5*t)*exp(-t)\n",
        "y2p = y2.diff(t)\n",
        "W = y1*y2p - y2*y1p\n",
        "W = simplify(W)\n",
        "W"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "22e4e064",
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        "id": "22e4e064"
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      "source": [
        "Obtain the particular solution using the formula\n",
        "\n",
        "$ y_p = -y1(x) \\int \\frac{y2(x) f(x)}{W(x)}dx + y2(x) \\int \\frac{y1(x) f(x)}{W(x)}dx $"
      ]
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    {
      "cell_type": "code",
      "execution_count": 6,
      "id": "7feca356",
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          "data": {
            "text/plain": [
              "4*sin(4*t) + 10*cos(2*t)**2 - 5"
            ],
            "text/latex": "$\\displaystyle 4 \\sin{\\left(4 t \\right)} + 10 \\cos^{2}{\\left(2 t \\right)} - 5$"
          },
          "metadata": {},
          "execution_count": 6
        }
      ],
      "source": [
        "Fx = 82*cos(4*t)\n",
        "yp1 = -y1* integrate(y2*Fx/W)\n",
        "yp2 = y2* integrate(y1*Fx/W)\n",
        "yp = yp1 + yp2\n",
        "simplify(yp)"
      ]
    },
    {
      "cell_type": "markdown",
      "id": "484b3598",
      "metadata": {
        "id": "484b3598"
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      "source": [
        "Note: The result can be simplified or rewritten either by rewriting sin(4t) or cos$^2$(2t)"
      ]
    },
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