What is the Laplace Transformation and Solution for C(X) = AX + B C_0 at x=0 and CL at x=L with concentration initially at C_0 between x=0 and L.

Step 1
Given:
$C(X)=AX+B\dots (i)$
We know that
$L(f(x))=F(s),L(x^n)=\frac{(n!)}{s^{n+1}},L(1)=\frac{1}{s}$
Taking Laplace transform in (i) using the above identities and the fact that it is linear.
$L(C(X))=AL(X)+BL(1)$
$L(C(s))=\frac{A}{s^2}+\frac{B}{s}\dots (ii)$
Step 2
Also, when x=0, from (i) we get
$C_0=B$ when x=L, from (i)
$C(L)=AL+B$
$CL=AL+C_0$
$A=\frac{C_L-C_0}{L}$ Putting value of A, B in (ii), we get
$L(C(s))=\frac{C_L-C_0}{L{s}^2}+\frac{C_0}{s}\dots (iii)$
On taking inverse Laplace Transform in (iii), we get
$C(X)=\frac{C_L-C_0}{L}X+C_0$