23.1.670 problem 665

Internal problem ID [5277]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 1. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF FIRST DEGREE, page 223
Problem number : 665
Date solved : Tuesday, September 30, 2025 at 12:06:40 PM
CAS classification : [_separable]

\begin{align*} x^{2} y^{2} y^{\prime }+1-x +x^{3}&=0 \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 111

ode:=x^2*y(x)^2*diff(y(x),x)+1-x+x^3 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {2^{{2}/{3}} {\left (-3 \left (x^{3}-\frac {2 c_1 x}{3}-2 \ln \left (x \right ) x -2\right ) x^{2}\right )}^{{1}/{3}}}{2 x} \\ y &= -\frac {2^{{2}/{3}} {\left (-3 \left (x^{3}-\frac {2 c_1 x}{3}-2 \ln \left (x \right ) x -2\right ) x^{2}\right )}^{{1}/{3}} \left (1+i \sqrt {3}\right )}{4 x} \\ y &= \frac {2^{{2}/{3}} {\left (-3 \left (x^{3}-\frac {2 c_1 x}{3}-2 \ln \left (x \right ) x -2\right ) x^{2}\right )}^{{1}/{3}} \left (i \sqrt {3}-1\right )}{4 x} \\ \end{align*}

✓ Mathematica. Time used: 0.159 (sec). Leaf size: 124

ode=x^2 y[x]^2 D[y[x],x]+1-x+x^3==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\sqrt [3]{-3} \sqrt [3]{\int _1^x\frac {-K[1]^3+K[1]-1}{K[1]^2}dK[1]+c_1}\\ y(x)&\to \sqrt [3]{3} \sqrt [3]{\int _1^x\frac {-K[1]^3+K[1]-1}{K[1]^2}dK[1]+c_1}\\ y(x)&\to (-1)^{2/3} \sqrt [3]{3} \sqrt [3]{\int _1^x\frac {-K[1]^3+K[1]-1}{K[1]^2}dK[1]+c_1} \end{align*}

✓ Sympy. Time used: 1.459 (sec). Leaf size: 104

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**3 + x**2*y(x)**2*Derivative(y(x), x) - x + 1,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = \frac {2^{\frac {2}{3}} \sqrt [3]{C_{1} - 3 x^{2} + 6 \log {\left (x \right )} + \frac {6}{x}}}{2}, \ y{\left (x \right )} = \frac {2^{\frac {2}{3}} \left (- \sqrt [3]{3} - 3^{\frac {5}{6}} i\right ) \sqrt [3]{C_{1} - x^{2} + 2 \log {\left (x \right )} + \frac {2}{x}}}{4}, \ y{\left (x \right )} = \frac {2^{\frac {2}{3}} \left (- \sqrt [3]{3} + 3^{\frac {5}{6}} i\right ) \sqrt [3]{C_{1} - x^{2} + 2 \log {\left (x \right )} + \frac {2}{x}}}{4}\right ] \]