I have some confusions about solutions of PDE. Let's say the short rate r(t) follows the CIR model:dr = k(m - r)dt + s*sqrt(r) dW,where k, m, s are positive constants. To find the value of a discount bond with maturity T, let its time-t value be D(t,T) or simply D(t). Some textbooks derive D(t) as follows:1) Write down the PDED_t + (s^2 r/2) D_{rr} + k(m-r) D_r - rD = 0.(here D_t means the partial derivative of D w.r.t. t and vice versa).2) Postulate that a solution is of the form D = exp(A(t) + B(t)r).3) Substitute the postulated form into the PDE, and solve it (some Ricatti equations stuff) with the terminal condition D(T)=1.Now suppose I want to find L(t,v) = E[exp(-v int_t^T r(u)du)] for some v>0. So it's some sort of Laplace transform and it's equal to D(t) when v=1.Question: shouldn't L also satisfy the same PDE and the same terminal condition as D does? If so, doesn't that mean D(t)=L(t,v)?!Thanks in advance.

Give me any system of pde's with unique solution f(x) [x a vector even] , now make up any new variable v and find a function g(x,v) such that g(x,1) = f(x). Then why would you expect g(x,v) = f(x) for all v ? Uniqueness no longer holds for functions of g(x,v) when you add the completely new variable v.