50.2.18 problem 4

Internal problem ID [7824]
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 : 4
Date solved : Thursday, April 03, 2025 at 01:21:37 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 \prime } y^{\prime }&=x \left (1+x \right ) \end{align*}

With initial conditions

\begin{align*} y \left (1\right )&=3 \end{align*}

✓ Maple. Time used: 0.094 (sec). Leaf size: 59

ode:=diff(diff(y(x),x),x)*diff(y(x),x) = x*(1+x); 
ic:=y(1) = 3; 
dsolve([ode,ic],y(x), singsol=all);
 

\begin{align*} y &= -\frac {\int _{1}^{x}\sqrt {6 \textit {\_z1}^{3}+9 \textit {\_z1}^{2}+9 c_1}d \textit {\_z1}}{3}+3 \\ y &= \frac {\int _{1}^{x}\sqrt {6 \textit {\_z1}^{3}+9 \textit {\_z1}^{2}+9 c_1}d \textit {\_z1}}{3}+3 \\ \end{align*}

✓ Mathematica. Time used: 61.314 (sec). Leaf size: 12885

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

Too large to display

✓ Sympy. Time used: 2.249 (sec). Leaf size: 44

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

\[ \left [ y{\left (x \right )} = C_{1} - \frac {\int \sqrt {C_{2} + 6 x^{3} + 9 x^{2}}\, dx}{3}, \ y{\left (x \right )} = C_{1} + \frac {\int \sqrt {C_{2} + 6 x^{3} + 9 x^{2}}\, dx}{3}\right ] \]