Scarlett Gauthier
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 title: Differential Equations 2
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 @@ -16,9 +15,6 @@ 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
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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 
 @@ -217,15 +213,12 @@ $$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
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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
 @@ -357,11 +350,7 @@ 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)
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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
 @@ -414,13 +403,11 @@ 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>
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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