diff --git a/src/8_differential_equations_2.md b/src/8_differential_equations_2.md
index b1034f92db4ab11d74e3628e7a91dfadaeb137e9..22f6d3c3d6b28314d8fae232dd5d7dfd06338df5 100644
--- a/src/8_differential_equations_2.md
+++ b/src/8_differential_equations_2.md
@@ -204,7 +204,7 @@ $$f(x) = e^{\lambda_1 x}, \ x e^{\lambda_1 x} , \ \cdots, \ x^{m_{1}-1} e^{\lamb
     since they are linear combinations of $f_1$ and $f_2$ which remain linearly
     independent,
     
-    $$\tilde{f_1}(x)=\cos(kx), \tilde{f_2}(x)=sin{kx}.$$
+    $$\tilde{f_1}(x)=\cos(kx), \tilde{f_2}(x)=\sin(kx).$$
     
     **Case 2: $E<0$**
     This time, define $E=-k^2$, for constant $k$. The characteristic polynomial 
@@ -487,9 +487,9 @@ the eigenfunctions of $L$.
 
 !!! info "Connection to quantum states"
     
-    Recall that q quantum state $\ket{\phi}$ can be written in an orthonormal 
-    basis $\{ \ket{u_n} \}$ as 
-    $$\ket{\phi} = \underset{n}{\Sigma} \bra{u_n} \ket{\phi} \ket{u_n}.$$ 
+    Recall that q quantum state $|\phi\rangle$ can be written in an orthonormal 
+    basis $\{ |u_n\rangle \}$ as 
+    $$\|\phi\rangle = \underset{n}{\Sigma} \langle u_n | \phi \rangle\, |u_n\rangle.$$ 
     
     In terms of hermitian operators and their eigenfunctions, the eigenfunctions
     play the role of the orthonormal basis. In reference to our running example,
@@ -500,7 +500,7 @@ To close our running example, consider the initial condition
 $\psi(x,o) = \psi_{0}(x)$. Since the eigenfunctions $\sin(\frac{n \pi x}{L})$ 
 form a basis, we can now write the general solution to the problem as 
 
-$$\psi(x,t)  = \overset{\infinity}{\underset{n}{\Sigma}} c_n e^{-i \frac{\lambda_n t}{\hbar}} \sin(\frac{n \pi x}{L}),$$
+$$\psi(x,t)  = \overset{\infty}{\underset{n}{\Sigma}} c_n e^{-i \frac{\lambda_n t}{\hbar}} \sin(\frac{n \pi x}{L}),$$
 
 where in the above we have defined the coefficients using the Fourier 
 coefficient,