4.2.49 $y^{\prime }(x)=\sqrt{|y(x)|}$

ODE
$$y^{\prime }(x)=\sqrt{|y(x)|}$$ ODE Classification

[_quadrature]

Book solution method
Separable ODE, Independent variable missing

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

$$\left\{\{y(x)\to \text{InverseFunction}[-\frac{22^{3/4}(1-\mathrm{\#}\text{1})\sqrt[4]{|\mathrm{\Im }(\mathrm{\#}\text{1})|+i(1-\mathrm{\Re }(\mathrm{\#}\text{1}))}(i|\mathrm{\Im }(\mathrm{\#}\text{1})|-\mathrm{\Re }(\mathrm{\#}\text{1})+1){}_2{F}_1(\frac{1}{4},\frac{3}{4};\frac{7}{4};\frac{2|\mathrm{\Im }(\mathrm{\#}\text{1})|+i({\mathrm{\#}\text{1}}^{\ast }+\mathrm{\#}\text{1}-2)}{4|\mathrm{\Im }(\mathrm{\#}\text{1})|})}{3\sqrt[4]{|\mathrm{\Im }(\mathrm{\#}\text{1})|}(\mathrm{\Im }(\mathrm{\#}\text{1}{)}^2+(1-\mathrm{\Re }(\mathrm{\#}\text{1}){)}^2)}-\frac{22^{3/4}(1-\mathrm{\#}\text{1})(-i|\mathrm{\Im }(\mathrm{\#}\text{1})|+\mathrm{\Im }(\mathrm{\#}\text{1}{)}^2+\mathrm{\Re }(\mathrm{\#}\text{1}{)}^2-\mathrm{\Re }(\mathrm{\#}\text{1}))\sqrt[4]{\frac{|\mathrm{\Im }(\mathrm{\#}\text{1})|+i((1-\mathrm{\Re }(\mathrm{\#}\text{1}))\mathrm{\Re }(\mathrm{\#}\text{1})-\mathrm{\Im }(\mathrm{\#}\text{1}{)}^2)}{\mathrm{\Im }(\mathrm{\#}\text{1}{)}^2+\mathrm{\Re }(\mathrm{\#}\text{1}{)}^2}}{}_2{F}_1(\frac{1}{4},\frac{3}{4};\frac{7}{4};\frac{2|\mathrm{\Im }(\mathrm{\#}\text{1})|+i(2{\left|\mathrm{\#}\text{1}\right|}^2-{\mathrm{\#}\text{1}}^{\ast }-\mathrm{\#}\text{1})}{4|\mathrm{\Im }(\mathrm{\#}\text{1})|})}{3\sqrt[4]{|\mathrm{\Im }(\mathrm{\#}\text{1})|}(\mathrm{\Im }(\mathrm{\#}\text{1}{)}^2+(1-\mathrm{\Re }(\mathrm{\#}\text{1}){)}^2)}\mathrm{\&}][c_1+x]\}\right\}$$

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

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