63.1.76 problem 121

Internal problem ID [15516]
Book : DIFFERENTIAL and INTEGRAL CALCULUS. VOL I. by N. PISKUNOV. MIR PUBLISHERS, Moscow 1969.
Section : Chapter 8. Differential equations. Exercises page 595
Problem number : 121
Date solved : Thursday, October 02, 2025 at 10:19:19 AM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

\begin{align*} y^{\prime \prime }&=\frac {a}{y^{3}} \end{align*}

✓ Maple. Time used: 0.149 (sec). Leaf size: 46

ode:=diff(diff(y(x),x),x) = a/y(x)^3; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 1.495 (sec). Leaf size: 63

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

\begin{align*} y(x)&\to -\frac {\sqrt {a+c_1{}^2 (x+c_2){}^2}}{\sqrt {c_1}}\\ y(x)&\to \frac {\sqrt {a+c_1{}^2 (x+c_2){}^2}}{\sqrt {c_1}}\\ y(x)&\to \text {Indeterminate} \end{align*}

✓ Sympy. Time used: 0.352 (sec). Leaf size: 112

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

\[ \left [ \begin {cases} \frac {i \sqrt {-1 + \frac {a}{C_{1} y^{2}{\left (x \right )}}} y{\left (x \right )}}{\sqrt {C_{1}}} & \text {for}\: \left |{\frac {a}{C_{1} y^{2}{\left (x \right )}}}\right | > 1 \\\frac {\sqrt {1 - \frac {a}{C_{1} y^{2}{\left (x \right )}}} y{\left (x \right )}}{\sqrt {C_{1}}} & \text {otherwise} \end {cases} = C_{1} + x, \ \begin {cases} \frac {i \sqrt {-1 + \frac {a}{C_{1} y^{2}{\left (x \right )}}} y{\left (x \right )}}{\sqrt {C_{1}}} & \text {for}\: \left |{\frac {a}{C_{1} y^{2}{\left (x \right )}}}\right | > 1 \\\frac {\sqrt {1 - \frac {a}{C_{1} y^{2}{\left (x \right )}}} y{\left (x \right )}}{\sqrt {C_{1}}} & \text {otherwise} \end {cases} = C_{1} - x\right ] \]