#!/usr/bin/env python
# coding: utf-8

# # Homework 21
# ## Symbolic math
# 
# ### Problem 1
# #### Part a
# 
# Import the library needed for using symbolic math in python. Also setup the notebook for printing.

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# #### Part b
# Set variables x, y, z, and function f, and g.

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# #### Part c
# 
# Set an expression for the following: $$x^2+2x-5.$$
# 

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#  #### Part d
#  
# 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
# #### Part a
# 
# Simplify the following expression:
# $$\frac{x^2 - x - 6}{x^2-3x}.$$

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# #### Part b
# 
# Expand the following expression symbolically: $$(x+1)^3(x-2)^2.$$

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# #### Part c
# Factor the following expression: $$3x^4 - 36x^3+99x^2-6x-144.$$

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# ### Problem 3
# #### Part a
# Compute the symbolic derivative: $$\frac{d}{dx}\sin^2(x)e^{2x}.$$
# Then evaluate the resulting expression for $x=3.3.$

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# #### Part b
# Create a sympy expression representing the following integral:
# $$\int_0^5x^2\sin(x^2)dx.$$
# 
# Then evaluate the integral symbolically.

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# ### Problem 4
# #### Part a
# Solve for the roots of the following equation: $$x^3+15x^2=3x-10.$$
# 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
# Solve the system of three equations in three unknowns symbolically:
# 
# $$x+y+z=0,$$
# 
# $$2x-y-z=10,$$
# 
# $$y+2z=5.$$
# 
# Compare the result to the answer computed with fsolve from scipy.optimize.

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# #### Part c
# Solve the following differential equation symbolically using the ```dsolve``` function:
# $$\frac{df(x)}{dx} = x\cos(x).$$

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# ### Problem 5
# #### Part a
# For the system $Ax=b$ with
# $$ A = \left[\begin{matrix}
# 1 & 2 & 5 \\
# 3 & 4 & 6 \\
# -1 & 0 & 3
# \end{matrix}\right],$$
# $$b = \left[\begin{matrix}
# 1 \\
# 0 \\
# -2
# \end{matrix}\right].$$
# Setup the matrices $A$ and $b$
# 

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# #### Part b
# For the system in Part a, solve for matrix $x$ by matrix algebra.

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# #### Part c
# 
# For matrix A above, return the middle row, and the middle column.

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