89.3.23 problem 24

Internal problem ID [24321]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 2. Equations of the first order and first degree. Exercises at page 34
Problem number : 24
Date solved : Thursday, October 02, 2025 at 10:17:56 PM
CAS classification : [_exact]

\begin{align*} \frac {1}{\left (1-y x \right )^{2}}+\left (y^{2}+\frac {x^{2}}{\left (1-y x \right )^{2}}\right ) y^{\prime }&=0 \end{align*}

With initial conditions

\begin{align*} y \left (4\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.940 (sec). Leaf size: 488

ode:=1/(1-x*y(x))^2+(y(x)^2+x^2/(1-x*y(x))^2)*diff(y(x),x) = 0; 
ic:=[y(4) = 1]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {1-\sqrt {\frac {2 \left (36 x^{3}+4 \sqrt {337 x^{6}+576 x^{5}+378 x^{4}+54 x^{3}+9 x^{2}+18 x +9}-12 x -12\right )^{{2}/{3}} x -32 x^{3}-24 x^{2}+\left (36 x^{3}+4 \sqrt {337 x^{6}+576 x^{5}+378 x^{4}+54 x^{3}+9 x^{2}+18 x +9}-12 x -12\right )^{{1}/{3}}}{\left (36 x^{3}+4 \sqrt {337 x^{6}+576 x^{5}+378 x^{4}+54 x^{3}+9 x^{2}+18 x +9}-12 x -12\right )^{{1}/{3}}}}+\sqrt {\frac {32 x^{3}+24 x^{2}-2 \left (36 x^{3}+4 \sqrt {337 x^{6}+576 x^{5}+378 x^{4}+54 x^{3}+9 x^{2}+18 x +9}-12 x -12\right )^{{2}/{3}} x +2 \left (36 x^{3}+4 \sqrt {337 x^{6}+576 x^{5}+378 x^{4}+54 x^{3}+9 x^{2}+18 x +9}-12 x -12\right )^{{1}/{3}}}{\left (36 x^{3}+4 \sqrt {337 x^{6}+576 x^{5}+378 x^{4}+54 x^{3}+9 x^{2}+18 x +9}-12 x -12\right )^{{1}/{3}}}+\frac {48 x^{3}-2}{\sqrt {\frac {2 \left (36 x^{3}+4 \sqrt {337 x^{6}+576 x^{5}+378 x^{4}+54 x^{3}+9 x^{2}+18 x +9}-12 x -12\right )^{{2}/{3}} x -32 x^{3}-24 x^{2}+\left (36 x^{3}+4 \sqrt {337 x^{6}+576 x^{5}+378 x^{4}+54 x^{3}+9 x^{2}+18 x +9}-12 x -12\right )^{{1}/{3}}}{\left (36 x^{3}+4 \sqrt {337 x^{6}+576 x^{5}+378 x^{4}+54 x^{3}+9 x^{2}+18 x +9}-12 x -12\right )^{{1}/{3}}}}}}}{4 x} \]

✗ Mathematica

ode=( 1-x*y[x] )^(-2)+ ( y[x]^2+x^2*( 1-x*y[x] )^(-2) )*D[y[x],x]==0; 
ic={y[4]==1}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Timed out

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((x**2/(-x*y(x) + 1)**2 + y(x)**2)*Derivative(y(x), x) + (-x*y(x) + 1)**(-2),0) 
ics = {y(4): 1} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out