DifferentialEquationsLecture2

Add lecture notes from the second lecture on differential equations.

src/8_differential_equations_2.md

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 title: Differential Equations 2
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 ## Higher order linear differential equations
 
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+# Higher order linear differential equations
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 In the previous lecture, we focused on first order linear differential equations
 as well as systems of such equations. In this lecture we switch focus to DE's 
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     $$f_1(x)=e^{0 x} = 1, \ f_{2}(x) = x e^{0 x} = x. $$
 
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 ## Partial differential equations: Separation of variables
 
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 ### Definitions and examples
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+# Partial differential equations
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 A partial differential equation (PDE) is an equation involving a function of two or 
 more indepenedent variables and derivatives of said function. These equations
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 in the figure below, regardless of the initial slope, the curves never reach $0$
 when $x=L$. 
 
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 ![image](figures/DE2_1.png)
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+<img src="figures/DE2_1.png" width="650">
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 For boundary value problems like this, there are only solutions for particular 
 eigenvalues $\lambda$. Coming back to the example, it turns out that solutions
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 These are still very special solutions. We will begin discussing next how to 
 obtain the general solution in our example. 
 
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 ## Self-adjoint differential operators
 
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+## Self-adjoint differential equations: Connection to Hilbert spaces! ##
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 As we hinted was possible earlier, let us re-write the previous equation by 
 defining a linear operator, $L$, acting on the space of functions which satisfy