77.35.12 problem 12

Internal problem ID [20716]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter VII. Exact differential equations and certain particular forms of equations. Misc. Exercise on chapter VII. Page 118
Problem number : 12
Date solved : Thursday, October 02, 2025 at 06:22:12 PM
CAS classification : [[_3rd_order, _missing_y], [_3rd_order, _with_linear_symmetries], [_3rd_order, _reducible, _mu_y2], [_3rd_order, _reducible, _mu_poly_yn]]

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

✓ Maple. Time used: 0.038 (sec). Leaf size: 61

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

\begin{align*} y &= \frac {4 \left (c_1 x +a^{2}\right )^{{5}/{2}}+15 c_1^{2} \left (c_2 x +c_3 \right )}{15 c_1^{2}} \\ y &= \frac {-4 \left (c_1 x +a^{2}\right )^{{5}/{2}}+15 c_1^{2} \left (c_2 x +c_3 \right )}{15 c_1^{2}} \\ \end{align*}

✓ Mathematica. Time used: 1.661 (sec). Leaf size: 75

ode=2*x*D[y[x],{x,3}]*D[y[x],{x,2}]==D[y[x],{x,2}]^2-a^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {4}{15} e^{-4 c_1} \left (a^2+e^{2 c_1} x\right ){}^{5/2}+c_3 x+c_2\\ y(x)&\to \frac {4}{15} e^{-4 c_1} \left (a^2+e^{2 c_1} x\right ){}^{5/2}+c_3 x+c_2 \end{align*}

✗ Sympy

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

Timed Out