50.2.17 problem 3

Internal problem ID [7823]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 1. What is a differential equation. Section 1.3 SEPARABLE EQUATIONS. Page 12
Problem number : 3
Date solved : Wednesday, March 05, 2025 at 05:07:16 AM
CAS classification : [[_2nd_order, _missing_y]]

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

✓ Maple. Time used: 0.010 (sec). Leaf size: 25

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

\[ y = -3 \Gamma \left (\frac {1}{3}, -\frac {x^{3}}{3}\right ) c_{2} \Gamma \left (\frac {2}{3}\right )+2 \pi \sqrt {3}\, c_{2} +c_{1} \]

✓ Mathematica. Time used: 0.035 (sec). Leaf size: 39

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

\[ y(x)\to \frac {c_1 \left (-x^3\right )^{2/3} \Gamma \left (\frac {1}{3},-\frac {x^3}{3}\right )}{3^{2/3} x^2}+c_2 \]

✓ Sympy. Time used: 0.398 (sec). Leaf size: 19

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

\[ y{\left (x \right )} = C_{1} + C_{2} \gamma \left (\frac {1}{3}, \frac {x^{3} e^{i \pi }}{3}\right ) \]