23.4.289 problem 292

Internal problem ID [6591]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 4. THE NONLINEAR EQUATION OF SECOND ORDER, page 380
Problem number : 292
Date solved : Friday, October 03, 2025 at 02:09:34 AM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

\begin{align*} {y^{\prime \prime }}^{2}&=a +b y \end{align*}

✓ Maple. Time used: 0.277 (sec). Leaf size: 206

ode:=diff(diff(y(x),x),x)^2 = a+b*y(x); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -\frac {a}{b} \\ b \sqrt {3}\, \int _{}^{y}\frac {1}{\sqrt {b \left (4 \textit {\_a} \sqrt {b \textit {\_a} +a}\, b +4 \sqrt {b \textit {\_a} +a}\, a -c_1 \right )}}d \textit {\_a} -x -c_2 &= 0 \\ -b \sqrt {3}\, \int _{}^{y}\frac {1}{\sqrt {b \left (4 \textit {\_a} \sqrt {b \textit {\_a} +a}\, b +4 \sqrt {b \textit {\_a} +a}\, a -c_1 \right )}}d \textit {\_a} -x -c_2 &= 0 \\ -b \sqrt {3}\, \int _{}^{y}\frac {1}{\sqrt {-b \left (4 \textit {\_a} \sqrt {b \textit {\_a} +a}\, b +4 \sqrt {b \textit {\_a} +a}\, a -c_1 \right )}}d \textit {\_a} -x -c_2 &= 0 \\ b \sqrt {3}\, \int _{}^{y}\frac {1}{\sqrt {-b \left (4 \textit {\_a} \sqrt {b \textit {\_a} +a}\, b +4 \sqrt {b \textit {\_a} +a}\, a -c_1 \right )}}d \textit {\_a} -x -c_2 &= 0 \\ \end{align*}

✓ Mathematica. Time used: 0.761 (sec). Leaf size: 201

ode=D[y[x],{x,2}]^2 == a + b*y[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} \text {Solve}\left [\frac {(a+b y(x))^2 \left (1-\frac {4 (a+b y(x))^{3/2}}{3 b c_1}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2}{3},\frac {5}{3},\frac {4 (a+b y(x))^{3/2}}{3 b c_1}\right ){}^2}{b^2 \left (-\frac {4 (a+b y(x))^{3/2}}{3 b}+c_1\right )}=(x+c_2){}^2,y(x)\right ]\\ \text {Solve}\left [\frac {(a+b y(x))^2 \left (1+\frac {4 (a+b y(x))^{3/2}}{3 b c_1}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2}{3},\frac {5}{3},-\frac {4 (a+b y(x))^{3/2}}{3 b c_1}\right ){}^2}{b^2 \left (\frac {4 (a+b y(x))^{3/2}}{3 b}+c_1\right )}=(x+c_2){}^2,y(x)\right ] \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
y = Function("y") 
ode = Eq(-a - b*y(x) + Derivative(y(x), (x, 2))**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : solve: Cannot solve -a - b*y(x) + Derivative(y(x), (x, 2))**2