23.4.277 problem 280

Internal problem ID [6579]
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 : 280
Date solved : Tuesday, September 30, 2025 at 03:10:59 PM
CAS classification : [[_2nd_order, _missing_y], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_poly_yn]]

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

✓ Maple. Time used: 0.012 (sec). Leaf size: 103

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

\begin{align*} y &= \frac {c_1 \ln \left (a x \,\operatorname {csgn}\left (a \right )+\sqrt {a^{2} x^{2}+c_1}\right ) \operatorname {csgn}\left (a \right )+a \left (x \sqrt {a^{2} x^{2}+c_1}+2 c_2 \right )}{2 a} \\ y &= \frac {-c_1 \ln \left (a x \,\operatorname {csgn}\left (a \right )+\sqrt {a^{2} x^{2}+c_1}\right ) \operatorname {csgn}\left (a \right )-a \left (x \sqrt {a^{2} x^{2}+c_1}-2 c_2 \right )}{2 a} \\ \end{align*}

✓ Mathematica. Time used: 13.687 (sec). Leaf size: 116

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

\begin{align*} y(x)&\to -\frac {1}{2} x \sqrt {a^2 x^2+2 c_1}-\frac {c_1 \log \left (a \left (a x+\sqrt {a^2 x^2+2 c_1}\right )\right )}{a}+c_2\\ y(x)&\to \frac {1}{2} x \sqrt {a^2 x^2+2 c_1}+\frac {c_1 \log \left (a \left (a x+\sqrt {a^2 x^2+2 c_1}\right )\right )}{a}+c_2 \end{align*}

✓ Sympy. Time used: 2.318 (sec). Leaf size: 82

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

\[ \left [ y{\left (x \right )} = C_{1} - \frac {\sqrt {C_{2}} x \sqrt {1 + \frac {a^{2} x^{2}}{C_{2}}}}{2} - \frac {C_{2} \operatorname {asinh}{\left (\frac {a x}{\sqrt {C_{2}}} \right )}}{2 a}, \ y{\left (x \right )} = C_{1} + \frac {\sqrt {C_{2}} x \sqrt {1 + \frac {a^{2} x^{2}}{C_{2}}}}{2} + \frac {C_{2} \operatorname {asinh}{\left (\frac {a x}{\sqrt {C_{2}}} \right )}}{2 a}\right ] \]