4.2.49 \(y'(x)=\sqrt {\left | y(x)\right | }\)

ODE
\[ y'(x)=\sqrt {\left | y(x)\right | } \] ODE Classification

[_quadrature]

Book solution method
Separable ODE, Independent variable missing

Mathematica
cpu = 95.6243 (sec), leaf count = 283

\[\left \{\left \{y(x)\to \text {InverseFunction}\left [-\frac {2\ 2^{3/4} (1-\text {$\#$1}) \sqrt [4]{\left | \Im (\text {$\#$1})\right | +i (1-\Re (\text {$\#$1}))} (i \left | \Im (\text {$\#$1})\right | -\Re (\text {$\#$1})+1) \, _2F_1\left (\frac {1}{4},\frac {3}{4};\frac {7}{4};\frac {2 \left | \Im (\text {$\#$1})\right | +i \left (\text {$\#$1}^*+\text {$\#$1}-2\right )}{4 \left | \Im (\text {$\#$1})\right | }\right )}{3 \sqrt [4]{\left | \Im (\text {$\#$1})\right | } \left (\Im (\text {$\#$1})^2+(1-\Re (\text {$\#$1}))^2\right )}-\frac {2\ 2^{3/4} (1-\text {$\#$1}) \left (-i \left | \Im (\text {$\#$1})\right | +\Im (\text {$\#$1})^2+\Re (\text {$\#$1})^2-\Re (\text {$\#$1})\right ) \sqrt [4]{\frac {\left | \Im (\text {$\#$1})\right | +i \left ((1-\Re (\text {$\#$1})) \Re (\text {$\#$1})-\Im (\text {$\#$1})^2\right )}{\Im (\text {$\#$1})^2+\Re (\text {$\#$1})^2}} \, _2F_1\left (\frac {1}{4},\frac {3}{4};\frac {7}{4};\frac {2 \left | \Im (\text {$\#$1})\right | +i \left (2 \left | \text {$\#$1}\right | ^2-\text {$\#$1}^*-\text {$\#$1}\right )}{4 \left | \Im (\text {$\#$1})\right | }\right )}{3 \sqrt [4]{\left | \Im (\text {$\#$1})\right | } \left (\Im (\text {$\#$1})^2+(1-\Re (\text {$\#$1}))^2\right )}\& \right ]\left [c_1+x\right ]\right \}\right \}\]

Maple
cpu = 0.168 (sec), leaf count = 31

\[ \left \{ x-\cases {-2\,\sqrt {-y \relax (x ) }&$y \relax (x ) \leq 0$\cr 2\,\sqrt {y \relax (x ) }&$0<y \relax (x ) $\cr }+{\it \_C1}=0 \right \} \] Mathematica raw input

DSolve[y'[x] == Sqrt[Abs[y[x]]],y[x],x]

Mathematica raw output

{{y[x] -> InverseFunction[(-2*2^(3/4)*Hypergeometric2F1[1/4, 3/4, 7/4, (2*Abs[Im
[#1]] + I*(-2 + Conjugate[#1] + #1))/(4*Abs[Im[#1]])]*(Abs[Im[#1]] + I*(1 - Re[#
1]))^(1/4)*(1 + I*Abs[Im[#1]] - Re[#1])*(1 - #1))/(3*Abs[Im[#1]]^(1/4)*(Im[#1]^2
 + (1 - Re[#1])^2)) - (2*2^(3/4)*Hypergeometric2F1[1/4, 3/4, 7/4, (2*Abs[Im[#1]]
 + I*(2*Abs[#1]^2 - Conjugate[#1] - #1))/(4*Abs[Im[#1]])]*((-I)*Abs[Im[#1]] + Im
[#1]^2 - Re[#1] + Re[#1]^2)*((Abs[Im[#1]] + I*(-Im[#1]^2 + (1 - Re[#1])*Re[#1]))
/(Im[#1]^2 + Re[#1]^2))^(1/4)*(1 - #1))/(3*Abs[Im[#1]]^(1/4)*(Im[#1]^2 + (1 - Re
[#1])^2)) & ][x + C[1]]}}

Maple raw input

dsolve(diff(y(x),x) = abs(y(x))^(1/2), y(x),'implicit')

Maple raw output

x-piecewise(y(x) <= 0,-2*(-y(x))^(1/2),0 < y(x),2*y(x)^(1/2))+_C1 = 0