ece204: add odes

docs/2a/ece205.md

@@ -225,6 +225,13 @@ Two boundary conditions are requred to solve the problem for all $t>0$ — that
 - $u(x,0)=f(x),0\leq x\leq L$
 - e.g., $u(0,t)=u(L,t)=0,t>0$
 
+Thus the general solution is:
+
+$$
+\boxed{u(x,t)=\sum^\infty_{n=1}a_ne^{-\left(\frac{n\pi a}{L}\right)^2t}\sin(\frac{n\pi x}{L})} \\
+f(x)=\sum^\infty_{n=1}a_n\sin(\frac{n\pi x}{L})
+$$
+
 ### Periodicity
 
 The **period** of a function is an increment that always returns the same value: $f(x+T)=f(x)$, and its **fundamental period** of a function is the smallest possible period.