Am I calculating the V(t) and I(t) of all of the elements in this RC circuit incorrectly? (two resitors and 1 cap connected to 10V DC)
There is a Thevenin equivalent that might help, somewhat. See here:







simulate this circuit &ndash; Schematic created using CircuitLab

The above, right is a very simple RC pair and it should be pretty easy for you to develop the equation for voltage and current with respect to the capacitor. If so, then at least you have your answer for node \$V\$. (Not the rest, but you are a lot closer to getting them, as well, too, as you can trivially develop anything else you need from there.)



The other approach is to just use nodal analysis:

$$\begin{align*}
\frac{V_t}{R_1}+\frac{V_t}{R_2}+C_1\frac{\text{d}V_t}{\text{d} t}&amp;=\frac{+10\:\text{V}}{R_1}+\frac{0\:\text{V}}{R_2}+C_1\frac{\text{d}\left(+10 \:\text{V}\right)}{\text{d} t}\\\\
\frac{V_t}{R_1}+\frac{V_t}{R_2}+C_1\frac{\text{d}V_t}{\text{d} t}&amp;=\frac{10\:\text{V}}{R_1}\\\\
\frac{V_t}{C_1\,R_1}+\frac{V_t}{C_1\,R_2}+\frac{\text{d}V_t}{\text{d} t}&amp;=\frac{10\:\text{V}}{C_1\,R_1}\\\\
\frac{\text{d}V_t}{\text{d} t}+\left(\frac{1}{C_1\,R_1}+\frac{1}{C_1\,R_2}\right)V_t&amp;=\frac{10\:\text{V}}{C_1\,R_1}
\end{align*}$$

That's just in standard form for the usual 1st order linear differential equation, which looks like this:

$$V_t^{'}+P_t\:V_t=Q_t$$

For convenience, set \$\tau_{_1}=R_1\cdot C_1\$, \$\tau_{_2}=R_2\cdot C_1\$, and \$\tau_{_0}=\frac{1}{\frac{1}{\tau_{_1}}+\frac{1}{\tau_{_2}}}=C_1\cdot\left(R_1\mid\mid R_2\right)\$.

If you want to see an exposition of how to use an integrating factor, \$\mu_{_t}\$, to solve the above then just look here, for example. There's plenty of explanation there. Here, set \$P_t=\frac{1}{\tau_{_0}}\$ and \$Q_t=10\:\text{V}\cdot \frac{1}{\tau_{_1}}\$ and find that \$\mu_{_t}=e^{^{\int P_t\:\text{d} t}}=e^{^\frac{t}{\tau_{_0}}}\$. \$Q_t\$ is a constant, so: \$V_t=\frac{1}{\mu_t} \int \mu_{_t}\:Q_t\:\text{d}t=\frac{Q_t}{\mu_t} \int \mu_{_t}\:\text{d}t\$. You should be able to solve from there, I think. You know the initial condition is \$V_0=10\:\text{V}\$ because the assumed voltage across \$C_1\$ is \$0\:\text{V}\$ (unless you know a different assumption to use there.)



Either approach gets you to the same place.