9.18.36 problem 36

Internal problem ID [3279]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 35, page 157
Problem number : 36
Date solved : Tuesday, September 30, 2025 at 06:32:05 AM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

\begin{align*} \left (1+{y^{\prime }}^{2}\right )^{2}&=y^{2} y^{\prime \prime } \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=3 \\ y^{\prime }\left (0\right )&=\sqrt {2} \\ \end{align*}

✓ Maple. Time used: 0.928 (sec). Leaf size: 221

ode:=(1+diff(y(x),x)^2)^2 = y(x)^2*diff(diff(y(x),x),x); 
ic:=[y(0) = 3, D(y)(0) = 2^(1/2)]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\begin{align*} y &= \operatorname {RootOf}\left (\sqrt {2}\, \int _{\textit {\_Z}}^{3}\frac {\operatorname {RootOf}\left (\left (-\sqrt {-\left (3 \textit {\_Z} -1\right ) \left (6 \textit {\_Z} +1\right )}+6 \textit {\_Z} -2\right ) \sqrt {2}\right ) \textit {\_a} -1}{\sqrt {-\left (\operatorname {RootOf}\left (\left (-\sqrt {-\left (3 \textit {\_Z} -1\right ) \left (6 \textit {\_Z} +1\right )}+6 \textit {\_Z} -2\right ) \sqrt {2}\right ) \textit {\_a} -1\right ) \left (2 \operatorname {RootOf}\left (\left (-\sqrt {-\left (3 \textit {\_Z} -1\right ) \left (6 \textit {\_Z} +1\right )}+6 \textit {\_Z} -2\right ) \sqrt {2}\right ) \textit {\_a} +\textit {\_a} -2\right )}}d \textit {\_a} +x \right ) \\ y &= \operatorname {RootOf}\left (\sqrt {2}\, \int _{3}^{\textit {\_Z}}\frac {\operatorname {RootOf}\left (\left (\sqrt {-\left (3 \textit {\_Z} -1\right ) \left (6 \textit {\_Z} +1\right )}+6 \textit {\_Z} -2\right ) \sqrt {2}\right ) \textit {\_a} -1}{\sqrt {-\left (\operatorname {RootOf}\left (\left (\sqrt {-\left (3 \textit {\_Z} -1\right ) \left (6 \textit {\_Z} +1\right )}+6 \textit {\_Z} -2\right ) \sqrt {2}\right ) \textit {\_a} -1\right ) \left (2 \operatorname {RootOf}\left (\left (\sqrt {-\left (3 \textit {\_Z} -1\right ) \left (6 \textit {\_Z} +1\right )}+6 \textit {\_Z} -2\right ) \sqrt {2}\right ) \textit {\_a} +\textit {\_a} -2\right )}}d \textit {\_a} +x \right ) \\ \end{align*}

✗ Mathematica

ode=(1+D[y[x],x]^2)^2==y[x]^2*D[y[x],{x,2}]; 
ic={y[0]==3,Derivative[1][y][0] ==Sqrt[2]}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Timed out

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((Derivative(y(x), x)**2 + 1)**2 - y(x)**2*Derivative(y(x), (x, 2)),0) 
ics = {y(0): 3, Subs(Derivative(y(x), x), x, 0): sqrt(2)} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -sqrt(y(x)*sqrt(Derivative(y(x), (x, 2))) - 1) + Derivative(y(x), x) cannot be solved by the factorable group method