2.1.5.2 Solved using first_order_ode_quadrature

Entering first order ode quadrature solver

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

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

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

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

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

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

Summary of solutions found

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