73.23.33 problem 33.11 (g)

Internal problem ID [15679]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 33. Power series solutions I: Basic computational methods. Additional Exercises. page 641
Problem number : 33.11 (g)
Date solved : Tuesday, January 28, 2025 at 08:04:04 AM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

\begin{align*} y^{\prime \prime }-y^{2}&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 52

Order:=5; 
dsolve(diff(y(x),x$2)-y(x)^2=0,y(x),type='series',x=0);
 

\[ y = \frac {x^{4} y \left (0\right )^{3}}{12}+\frac {y \left (0\right )^{2} x^{2}}{2}+\left (1+\frac {y^{\prime }\left (0\right ) x^{3}}{3}\right ) y \left (0\right )+y^{\prime }\left (0\right ) x +\frac {{y^{\prime }\left (0\right )}^{2} x^{4}}{12}+O\left (x^{5}\right ) \]

✓ Solution by Mathematica

Time used: 0.025 (sec). Leaf size: 147

AsymptoticDSolveValue[D[y[x],{x,2}]-y[x]^2==0,y[x],{x,0,"5"-1}]
 

\[ y(x)\to \frac {1}{12} x^4 \left (6 \wp \left (\frac {c_1}{\sqrt [3]{6}};0,c_2\right ){}^3+\wp ''\left (\frac {c_1}{\sqrt [3]{6}};0,c_2\right ){}^2\right )+\frac {\sqrt [3]{2} x^3 \wp \left (\frac {c_1}{\sqrt [3]{6}};0,c_2\right ) \wp ''\left (\frac {c_1}{\sqrt [3]{6}};0,c_2\right )}{3^{2/3}}+\frac {3^{2/3} x^2 \wp \left (\frac {c_1}{\sqrt [3]{6}};0,c_2\right ){}^2}{\sqrt [3]{2}}+\sqrt [3]{6} \wp \left (\frac {c_1}{\sqrt [3]{6}};0,c_2\right )+x \wp ''\left (\frac {c_1}{\sqrt [3]{6}};0,c_2\right ) \]