23.4.66 problem 66

Internal problem ID [6368]
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 : 66
Date solved : Tuesday, September 30, 2025 at 02:53:29 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

\begin{align*} y^{\prime \prime }&=a y {\left (1+\left (b -y^{\prime }\right )^{2}\right )}^{{3}/{2}} \end{align*}

✓ Maple. Time used: 0.365 (sec). Leaf size: 224

ode:=diff(diff(y(x),x),x) = a*y(x)*(1+(b-diff(y(x),x))^2)^(3/2); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \left (b -i\right ) x +c_1 \\ y &= \left (b +i\right ) x +c_1 \\ b \int _{}^{y}\frac {\left (\textit {\_a}^{2}+2 c_1 \right )^{2} a^{2}-4 b^{2}}{\left (\textit {\_a}^{2}+2 c_1 \right ) a \sqrt {-\left (\left (\textit {\_a}^{2}+2 c_1 \right )^{2} a^{2}-4 b^{2}-4\right ) b^{2}}+\left (\left (\textit {\_a}^{2}+2 c_1 \right )^{2} a^{2}-4 b^{2}-4\right ) b^{2}}d \textit {\_a} -x -c_2 &= 0 \\ -b \int _{}^{y}\frac {-\left (\textit {\_a}^{2}+2 c_1 \right )^{2} a^{2}+4 b^{2}}{-\left (\textit {\_a}^{2}+2 c_1 \right ) a \sqrt {-\left (\left (\textit {\_a}^{2}+2 c_1 \right )^{2} a^{2}-4 b^{2}-4\right ) b^{2}}+\left (\left (\textit {\_a}^{2}+2 c_1 \right )^{2} a^{2}-4 b^{2}-4\right ) b^{2}}d \textit {\_a} -x -c_2 &= 0 \\ \end{align*}

✓ Mathematica. Time used: 64.206 (sec). Leaf size: 1391

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

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✗ Sympy

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

NotImplementedError : The given ODE -b - sqrt(2**(2/3)*(sqrt(Derivative(y(x), (x, 2))**4/(a**4*y(x)*