Merge branch 'DE2' of...

Merge branch 'DE2' of ssh://gitlab.kwant-project.org:443/mathematics-for-quantum-physics/lectures into DE2

src/8_differential_equations_2.md

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 title: Differential Equations 2
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@@ -15,6 +16,9 @@ The lecture on differential equations consists of three parts, each with their o
 ## Higher order linear differential equations
 
 <iframe width="100%" height=315 src="https://www.youtube-nocookie.com/embed/ucvIiLgJ2i0?rel=0" frameborder="0" allow="accelerometer; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>
+=======
+# Higher order linear differential equations
+>>>>>>> df33d26965bff04a2e2724e3894837eddf801f3a
 
 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 
@@ -213,11 +217,15 @@ $$f(x) = e^{\lambda_1 x}, \ x e^{\lambda_1 x} , \ \cdots, \ x^{m_{1}-1} e^{\lamb
     
     $$f_1(x)=e^{0 x} = 1, \ f_{2}(x) = x e^{0 x} = x. $$
 
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 ## Partial differential equations: Separation of variables
 
 <iframe width="100%" height=315 src="https://www.youtube-nocookie.com/embed/I4ghpYsFLFY?rel=0" frameborder="0" allow="accelerometer; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>
 
 ### Definitions and examples
+=======
+# Partial differential equations
+>>>>>>> df33d26965bff04a2e2724e3894837eddf801f3a
 
 A partial differential equation (PDE) is an equation involving a function of two or 
 more indepenedent variables and derivatives of said function. These equations
@@ -349,7 +357,11 @@ problem, a boundary value problem does not always have a solution. For example,
 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)
+=======
+<img src="figures/DE2_1.png" width="650">
+>>>>>>> df33d26965bff04a2e2724e3894837eddf801f3a
 
 For boundary value problems like this, there are only solutions for particular 
 eigenvalues $\lambda$. Coming back to the example, it turns out that solutions
@@ -402,9 +414,13 @@ Notice that there is one solution $\psi_{n}(x,t)$ for each natural number $n$.
 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
 
 <iframe width="100%" height=315 src="https://www.youtube-nocookie.com/embed/p4MHW0yMMvY?rel=0" frameborder="0" allow="accelerometer; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>
+=======
+## Self-adjoint differential equations: Connection to Hilbert spaces! ##
+>>>>>>> df33d26965bff04a2e2724e3894837eddf801f3a
 
 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