diff --git a/src/8_differential_equations_2.md b/src/8_differential_equations_2.md index a94a60d8136b66e0ebcb8292c0c64eb91dc0de8a..64830f358f0299a0de9e069a659e0ffe284c09a3 100644 --- a/src/8_differential_equations_2.md +++ b/src/8_differential_equations_2.md @@ -198,7 +198,7 @@ $$f(x) = e^{\lambda_1 x}, \ x e^{\lambda_1 x} , \ \cdots, \ x^{m_{1}-1} e^{\lamb since they are linear combinations of $f_1$ and $f_2$ which remain linearly independent, - $$\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 @@ -397,9 +397,9 @@ In the previous equation, the coefficient $A$ can be determined if the original PDE was supplied with an initial condition. Putting the solutions to the two ODEs together and redefining -$\Tilde{A}=A \cdot c_1$, we arrive at the solutions for theb PDE, +$\tilde{A}=A \cdot c_1$, we arrive at the solutions for theb PDE, -$\psi_n(x,t) = \Tilde{A}_n e^{-i \frac{\lambda_n t}{\hbar}} sin(\frac{n \pi x}{L}).$ +$\psi_n(x,t) = \tilde{A}_n e^{-i \frac{\lambda_n t}{\hbar}} sin(\frac{n \pi x}{L}).$ 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