2.1.6.2 Solved using first_order_ode_quadrature

Entering first order ode quadrature solver

\[\begin {aligned} x^{\prime }&=\frac {\cos \left (t \right )}{\sin \left (t \right )}\\ x \left (1\right ) &= 0\\ \end {aligned}\]

Because the ODE has the form \(x^{\prime }=f(t)\), the solution requires only integration. Therefore

\begin{align*} dx &= \left (\frac {\cos \left (t \right )}{\sin \left (t \right )}\right ) \, dt\\ x &= \int { \left (\frac {\cos \left (t \right )}{\sin \left (t \right )}\right ) \, dt}\\ &= \ln \left (\sin \left (t \right )\right )+c_1 \end{align*}

Applying the initial condition \(x \left (1\right ) = 0\), the solution becomes

\begin{align*} x &= \ln \left (\sin \left (t \right )\right )-\ln \left (\sin \left (1\right )\right ) \\ \end{align*}

Direction field with Isoclines	\( x = \ln \left (\sin \left (t \right )\right )-\ln \left (\sin \left (1\right )\right ) \)

Summary of solutions found

\[ x = \ln \left (\sin \left (t \right )\right )-\ln \left (\sin \left (1\right )\right ) \]