# Need do a Laplace transformation for T_1 * V′(t)+V(t)=K * P(t)+Va

I am trying to do a Laplace transform of the following equation:
T1⋅V′(t)+V(t)=K⋅P(t)+Va
The purpose is to solve V(s) equation knowing that V(0)=Va. I followed all the transformation rules and I got the following result:
V(s)=K⋅P(s)s⋅(1+T1)+Vas
But, according to the correction, the expected result is :
V(s)=K⋅P(s)(1+s⋅T1)+Vas
What have I done wrong?
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Solomon Fernandez
${T}_{1}\cdot {V}^{\prime }\left(t\right)+V\left(t\right)=K\cdot P\left(t\right)+Va$
Apply Laplace Transform:
${T}_{1}\left(sV\left(s\right)-V\left(0\right)\right)+V\left(s\right)=K\cdot P\left(S\right)+\frac{Va}{s}$
$V\left(s\right)\left(s{T}_{1}+1\right)=K\cdot P\left(S\right)+\frac{Va}{s}+{T}_{1}V\left(0\right)$
Since $V\left(0\right)=Va$ we have:
$V\left(s\right)\left(s{T}_{1}+1\right)=K\cdot P\left(S\right)+\frac{Va}{s}+{T}_{1}Va$
$V\left(s\right)\left(s{T}_{1}+1\right)=K\cdot P\left(S\right)+\frac{Va\left(1+s{T}_{1}\right)}{s}$
It finally gives:
$V\left(s\right)=K\cdot \frac{P\left(S\right)}{\left(s{T}_{1}+1\right)}+\frac{Va}{s}$