<s>[INST] Solve the differential equation $\frac{dx}{dt}=-\frac{1}{5}\sqrt{(x+1)}$ Solve the differential equation $$\frac{dx}{dt}=-\frac{1}{5}\sqrt{(x+1)}$$ given that $x=80$ when $t=0$. Give your answer in the form $x=f(t)$.
\begin{align}
\frac{dx}{dt} & = -\frac{1}{5}\sqrt{(x+1)} \\
\frac{dt}{dx} & = \frac{-5}{\sqrt{(x+1)}} \\ 
\int-\frac{1}{5}dt & = \int\frac{1}{\sqrt{(x+1)}}dx \\
-\frac{1}{5}t & = 2\sqrt{(x+1)} \\ 
-\frac{t}{10} & = \sqrt{(x+1)} \\ 
\left(-\frac{t}{10}\right)^2 & = x+1 \\ 
x & = \left(-\frac{t}{10}\right)^2-1+C \\ 
\end{align}
Let $x=80$ and $t=0$
\begin{align}
80 & = -1+C \\
C & = 81 \\ 
\end{align}
Answer in the Mark Scheme is $C=18$
$$x=\left(9-\frac{t}{10}\right)^2-1$$
 [/INST] You didn't put $+C$ in the right place. You should have $C-\frac{t}{10}=\sqrt{x+1}$. Solve for $C$ here, then isolate $x$.
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