Chapter 10 - Introduction to Differential Equations - 10.1 Solving Differential Equations - Exercises - Page 505: 30

$$x =\sin ^{-1}\left(C e^{t / 2}\right) $$

\begin{aligned} \frac{dx}{dt}&=t\tan t\\ \cot x d x &=t d t \\ \int \cot x d x &=\int t d t \\ \ln \sin x &=\frac{t^{2}}{2}+\ln C \\ \sin x &=C e^{t^{2} / 2} \\ x &=\sin ^{-1}\left(C e^{t / 2}\right) \end{aligned}