# 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.

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