2.1.11.7 ✓ Maple. Time used: 0.060 (sec). Leaf size: 9

ode:=diff(x(t),t) = 2*x(t)^(1/2); 
ic:=[x(0) = 1]; 
dsolve([ode,op(ic)],x(t), singsol=all);
 

\begin{align*} x = \left (t +1\right )^{2} \end{align*}

Maple trace

Methods for first order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
trying 1st order linear 
trying Bernoulli 
<- Bernoulli successful
 

Maple step by step

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [\frac {d}{d t}x \left (t \right )=2 \sqrt {x \left (t \right )}, x \left (0\right )=1\right ] \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d t}x \left (t \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d t}x \left (t \right )=2 \sqrt {x \left (t \right )} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {\frac {d}{d t}x \left (t \right )}{\sqrt {x \left (t \right )}}=2 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int \frac {\frac {d}{d t}x \left (t \right )}{\sqrt {x \left (t \right )}}d t =\int 2d t +\mathit {C1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & 2 \sqrt {x \left (t \right )}=2 t +\mathit {C1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} x \left (t \right ) \\ {} & {} & x \left (t \right )=t^{2}+t \mathit {C1} +\frac {1}{4} \mathit {C1}^{2} \\ \bullet & {} & \textrm {Simplify}\hspace {3pt} \\ {} & {} & x \left (t \right )=\left (t +\frac {\mathit {C1}}{2}\right )^{2} \\ \bullet & {} & \textrm {Redefine the integration constant(s)}\hspace {3pt} \\ {} & {} & x \left (t \right )=\left (t +\mathit {C1} \right )^{2} \\ \bullet & {} & \textrm {Use initial condition}\hspace {3pt} x \left (0\right )=1 \\ {} & {} & 1=\mathit {C1}^{2} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} \textit {\_C1} \\ {} & {} & \left \{\mathit {C1} =-1, \mathit {C1} =1\right \} \\ \bullet & {} & \mathrm {Remove solutions that don\esapos t satisfy the ODE} \\ {} & {} & \mathit {C1} =1 \\ \bullet & {} & \textrm {Substitute}\hspace {3pt} \textit {\_C1} =1\hspace {3pt}\textrm {into general solution and simplify}\hspace {3pt} \\ {} & {} & x \left (t \right )=\left (t +1\right )^{2} \\ \bullet & {} & \textrm {Solution to the IVP}\hspace {3pt} \\ {} & {} & x \left (t \right )=\left (t +1\right )^{2} \end {array} \]