Consider the following transient PDE:

$\frac{\text{d}\text{e}\text{l}\text{u}}{\text{d}\text{e}\text{l}\text{t}}=v\frac{de{l}^{2}u}{del{y}^2}$

with similarity transformations in the form:

$=\frac{y}{2\sqrt[\phantom{2}]{vt}},$ and $,u(y,t)=Uf()$ where $$ is a nondimensional similarity

variable and $v$ and $U$ are constant kinematic viscosity and freestream velocity values,

respectively.

a$)$ Classify the given PDE.

b$)$ Derive $\frac{\text{d}\text{e}\text{l}\text{u}}{\text{d}\text{e}\text{l}\text{t}},\frac{\text{d}\text{e}\text{l}\text{u}}{\text{d}\text{e}\text{l}\text{y}},$ and $\frac{de{l}^{2}u}{del{y}^2}$ in terms of the similarity variable $.$

c$)$ Derive the resulting differential equation in terms of $f()$ and its derivatives.

d$)$ Discuss the properties of the obtained differential equation.

Please solve each question. Please show all intermadiate steps. Please solving on paper with deep explanation.$($need all steps$)$