HW17.ipynb

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    "# Homework 17\n",
    "## Symbolic math\n",
    "\n",
    "### Problem 1\n",
    "#### Part a\n",
    "\n",
    "Import the library needed for using symbolic math in python. Also setup the notebook for printing."
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    "#### Part b\n",
    "Set variables x, y, z, and function f, and g."
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    "#### Part c\n",
    "\n",
    "Set an expression for the following: $$x^2+2x-5.$$\n"
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    " #### Part d\n",
    " \n",
    "Evaluate the expression for $x=1.5$. Also, make a variable substitution: $z$ for $x$. Do a variable substitution $y^2$ for x. "
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    "### Problem 2\n",
    "#### Part a\n",
    "\n",
    "Simplify the following expression:\n",
    "$$\\frac{x^2 - x - 6}{x^2-3x}.$$"
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    "#### Part b\n",
    "\n",
    "Expand the following expression symbolically: $$(x+1)^3(x-2)^2.$$"
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    "#### Part c\n",
    "Factor the following expression: $$3x^4 - 36x^3+99x^2-6x-144.$$"
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    "### Problem 3\n",
    "#### Part a\n",
    "Compute the symbolic derivative: $$\\frac{d}{dx}\\sin^2(x)e^{2x}.$$\n",
    "Then evaluate the resulting expression for $x=3.3.$"
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    "#### Part b\n",
    "Create a sympy expression representing the following integral:\n",
    "$$\\int_0^5x^2\\sin(x^2)dx.$$\n",
    "\n",
    "Then evaluate the integral symbolically."
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    "### Problem 4\n",
    "#### Part a\n",
    "Solve for the roots of the following equation: $$x^3+15x^2=3x-10.$$\n",
    "Use the ```Eq``` and ```solve``` functions and save as an expression. Show the expression (it will be a list). Then find the numerical value of each root using the evalf function. You can use evalf on some expression using ```my_expression.evalf()```."
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    "#### Part b\n",
    "Solve the system of three equations in three unknowns symbolically:\n",
    "\\begin{align}\n",
    "x+y+z&=0 \\\\\n",
    "2x-y-z&=10 \\\\\n",
    "y+2z&=5\n",
    "\\end{align}\n",
    "Compare the result to the answer computed with fsolve from scipy.optimize."
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    "#### Part c\n",
    "Solve the following differential equation symbolically using the ```dsolve``` function:\n",
    "$$\\frac{df(x)}{dx} = x\\cos(x).$$"
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    "### Problem 5\n",
    "#### Part a\n",
    "For the system $Ax=b$ with\n",
    "$$ A = \\left[\\begin{matrix}\n",
    "1 & 2 & 5 \\\\\n",
    "3 & 4 & 6 \\\\\n",
    "-1 & 0 & 3\n",
    "\\end{matrix}\\right],$$\n",
    "$$b = \\left[\\begin{matrix}\n",
    "1 \\\\\n",
    "0 \\\\\n",
    "-2\n",
    "\\end{matrix}\\right].$$\n",
    "Setup the matrices $A$ and $b$\n"
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    "#### Part b\n",
    "For the system in Part a, solve for matrix $x$ by matrix algebra."
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    "#### Part c\n",
    "\n",
    "For matrix A above, return the middle row, and the middle column."
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    "#### Part d\n",
    "Create a matrix $M$ using the ```zeros``` function that has 2 rows and 2 columns. Fill in some values using array notation (like M[i,j]=value)."
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