General solution to partial differential equation (PDE)

[manchester20]

Hi,
I have the following PDE$-S\frac{\partial\vartheta}{\partial\tau}+\frac{1}{2}\sigma^2\frac{X^2}{S}\frac{\partial^2\vartheta}{\partial\xi^{2}} + [\frac{S}{T} + (r-D)X]\frac{\partial\vartheta}{\partial\xi}$I am asked to seek a solution of the form $\vartheta=\alpha_1(\tau)\xi + \alpha_0(\tau)$ and give a general solution for $\alpha_1(\tau)$ and $\alpha_0(\tau)$

where we have
$\tau=T-t$
and
$\xi=\frac{t}{T}-\frac{X}{S}$

I have tried doing the partial differentials of $\vartheta$ with respect to τ and ε, but the answer doesn't allow me to get a general solution for the two unknown functions of τ.
If anyone could help i would be really grateful.
Thanks

NOTE: the word 'partial' in the equation should be a symbol for the partial derivative.

 

[JJacquelin]

Hi,
I have the following PDE


$-S\frac{\partial\vartheta}{\partial\tau}+\frac{1}{2}\sigma^2\frac{X^2}{S}\frac{\partial^2\vartheta}{\partial\xi^{2}} + [\frac{S}{T} + (r-D)X]\frac{\partial\vartheta}{\partial\xi}$


I am asked to seek a solution of the form $\vartheta=\alpha_1(\tau)\xi + \alpha_0(\tau)$ and give a general solution for $\alpha_1(\tau)$ and $\alpha_0(\tau)$

where we have
$\tau=T-t$
and
$\xi=\frac{t}{T}-\frac{X}{S}$

I have tried doing the partial differentials of $\vartheta$ with respect to τ and ε, but the answer doesn't allow me to get a general solution for the two unknown functions of τ.
If anyone could help i would be really grateful.
Thanks

NOTE: the word 'partial' in the equation should be a symbol for the partial derivative.

Hi !

Sorry to say, but the wording of the problem seems very fishy (see attachment)