2.1.20.1 Solved using first_order_ode_linear

Entering first order ode linear solver

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

In canonical form a linear first order is

\begin{align*} x^{\prime } + q(t)x &= p(t) \end{align*}

Comparing the above to the given ode shows that

\begin{align*} q(t) &=\tan \left (t \right )\\ p(t) &=0 \end{align*}

The integrating factor \(\mu \) is

\begin{align*} \mu &= e^{\int {q\,dt}}\\ &= {\mathrm e}^{\int \tan \left (t \right )d t}\\ &= \sec \left (t \right ) \end{align*}

The ode becomes

\begin{align*} \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}t}} \mu x &= 0 \\ \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}t}} \left (x \sec \left (t \right )\right ) &= 0 \end{align*}

Integrating gives

\begin{align*} x \sec \left (t \right )&= \int {0 \,dt} + c_1 \\ &=c_1 \end{align*}

Dividing throughout by the integrating factor \(\sec \left (t \right )\) gives the final solution

\[ x = c_1 \cos \left (t \right ) \]

Direction field with Isoclines	Direction field

Summary of solutions found

\[ x = c_1 \cos \left (t \right ) \]