Replace y[L] ==0 with y'[0] == yp0 and then combine y[L] ==0 with the solution you get.

The solution given is correct but it isn't the only solution. Yes, you'll have to handle the boundary condition at L separately.

I recommend not using capital letters for your constants (or at least give them unique names) because for example I is the symbol for Sqrt[-1] and E is the euler number Here is how I would solve it:

In[13]:= Clear["Global`*"]

In[14]:= sol = 
 DSolve[{y''[x] + ps/(es is) y[x] == 0, y[0] == 0}, y[x], x]

Out[14]= {{y[x] -> C[2] Sin[(Sqrt[ps] x)/(Sqrt[es] Sqrt[is])]}}

In[15]:= Solve[Sqrt[ps] l/Sqrt[es is] == n Pi, ps]

Out[15]= {{ps -> (es is n^2 \[Pi]^2)/l^2}}