2.1.11.2 Solved using first_order_ode_autonomous
Entering first order ode autonomous solver
\[\begin {aligned} x^{\prime }&=2 \sqrt {x}\\ x \left (0\right ) &= 1\\ \end {aligned}\]
Integrating gives
\begin{align*} \int \frac {1}{2 \sqrt {x}}d x &= dt\\ \sqrt {x}&= t +c_1 \end{align*}
Applying the initial condition \(x \left (0\right ) = 1\), the solution becomes
\begin{align*}
\sqrt {x} &= t +1 \\
\end{align*}
Solving for \(x\) from \(\sqrt {x} = t +1\) gives
\begin{align*}
x &= t^{2}+2 t +1 \\
\end{align*}
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| Direction field with Isoclines | \( x = t^{2}+2 t +1 \) |
Summary of solutions found
\[
x = t^{2}+2 t +1
\]