I am trying to understand why I'm not getting the same answer of using a certain method for solving circuits. For the following example, the switch opens at t = 0, I am to find the capacitor voltage at t>0.

The first method I tried was to write the differential equation for the capacitor voltage and find the LaPlace transform of it :

\(frac{1}{2}frac{d^2v}{dt}+frac{3}{2}frac{dv}{dt}+v=\0\) => \(V(s)=\frac{12s+28}{3(s+1)(s+2)\) => \(v(t)=frac{16}{3}e^(-t)-frac{4}{3}e^(-2t)\)

However, representing it in the s-domain leads me to another equation:

\(V(s)=frac{-8s-24}{3s(s+1)(s+2)\) => \(v(t)=[frac{16}{3}e^(-t)-frac{4}{3}e^(-2t)-4]u(t)\)

Moderators note : removed the plain tags to show the latex

The first method I tried was to write the differential equation for the capacitor voltage and find the LaPlace transform of it :

\(frac{1}{2}frac{d^2v}{dt}+frac{3}{2}frac{dv}{dt}+v=\0\) => \(V(s)=\frac{12s+28}{3(s+1)(s+2)\) => \(v(t)=frac{16}{3}e^(-t)-frac{4}{3}e^(-2t)\)

However, representing it in the s-domain leads me to another equation:

\(V(s)=frac{-8s-24}{3s(s+1)(s+2)\) => \(v(t)=[frac{16}{3}e^(-t)-frac{4}{3}e^(-2t)-4]u(t)\)

Moderators note : removed the plain tags to show the latex

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