2.6.9.1 Solved using first_order_ode_quadrature

Entering first order ode quadrature solver

\[\begin {aligned} \sqrt {1+v^{\prime }}&=\frac {{\mathrm e}^{u}}{2} \end {aligned}\]

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

\begin{align*} dv &= \left (\frac {{\mathrm e}^{2 u}}{4}-1\right ) \, du\\ v &= \int { \left (\frac {{\mathrm e}^{2 u}}{4}-1\right ) \, du}\\ &= -u +\frac {{\mathrm e}^{2 u}}{8}+c_1 \end{align*}

Direction field with Isoclines	Direction field

Summary of solutions found

\[ v = -u +\frac {{\mathrm e}^{2 u}}{8}+c_1 \]