Cannot perform Laplace transformation to integrate ODE correctly Given

*

*two state (time-dependent) parameters: $C,A\ge0$

*these fixed parameters: $R,k_0,V,k_1,k_2\gt0$
and the following two differential equations
$\begin{cases}
\frac{dC}{dt}=\frac{R}{V}-(k_1+k_0)C+\frac{k_2A}{V}\\
\frac{dA}{dt}=k_1CV-k_2A
 \end{cases}$
I would like to integrate the above and obtain the analytical solution for $C(t)$ for any set of non-negative initial conditions: $C(0)$ and $A(0)$.
I adopted the technique of Laplace transformation as introduced in this article.
The followings are the general step that I have followed:

Applying Laplace transformation to domain $s$ for $A$:
$s{\cdot}A(s)-A(0)=k_1VC(s)-k_2A(s)$
$A(s)=\frac{k_1VC(s)+A(0)}{s+k_2}$  --------------------------------------- (1)
Applying Laplace transformation to domain $s$ for $C$:
$s{\cdot}C(s)-C(0)=\frac{R}{V}-(k_1+k_0)C(s)+\frac{k_2A(s)}{V}$
By substituting equation (1), followed by some simplification, we have:
$C(s)=\frac{R+VC(0)}{V}{\cdot}\frac{s+k_2+\frac{k_2A(0)}{R+VC(0)}}{s^2+s{\cdot}k_1+s{\cdot}k_0+s{\cdot}k_2+k_2{\cdot}k_0}$
Then, the denominator can be simplified with the quadratic formula:
$\begin{cases}
C(s)=\frac{R+VC(0)}{V}{\cdot}\frac{s+k_2+\frac{k_2A(0)}{R+VC(0)}}{(s+\alpha)(s+\beta)}\\
\alpha,\beta=\frac{k_1+k_2+k_0\pm\sqrt{(k_1+k_2+k_0)^2-4k_2{\cdot}k_0}}{2}
\end{cases}$
This form contains a constant term times a proper polynomial in $s$ (i.e. degree in the numerator is smaller than the denominator), so it can be converted back to the time domain by inverse Laplace transform:
$C(t)=\frac{R+VC(0)}{V}\left(\frac{-\alpha+k_2+\frac{k_2A(0)}{R+VC(0)}}{\beta-\alpha}e^{-\alpha t}+\frac{-\beta+k_2+\frac{k_2A(0)}{R+VC(0)}}{\alpha-\beta}e^{-\beta t}\right)$

The problem is, I verified my solution against a differential equation solver and they gave very different numbers... So I suspect that there are some errors with my integration, but I cannot identify the error...  Can anyone please help check my answer?

The Laplace transform of $R/V_p$ is $R/(sV_p)$. Does that fix your problem?
