69.7.23 problem 198

Internal problem ID [18103]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 7, Total differential equations. The integrating factor. Exercises page 61
Problem number : 198
Date solved : Thursday, October 02, 2025 at 02:42:20 PM
CAS classification : [_rational, [_1st_order, `_with_symmetry_[F(x)*G(y),0]`], [_Abel, `2nd type`, `class A`]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 55

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

\begin{align*} y &= \frac {2 c_1 +1-\sqrt {2 c_1 \,x^{2}+2 c_1 +1}}{2 c_1} \\ y &= \frac {2 c_1 +1+\sqrt {2 c_1 \,x^{2}+2 c_1 +1}}{2 c_1} \\ \end{align*}

✓ Mathematica. Time used: 3.248 (sec). Leaf size: 295

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

\begin{align*} y(x)&\to -x^2+\frac {1}{\frac {1}{x^2+1}-\frac {1+i}{\left (x^2+1\right ) \sqrt {-2 \left (x^2+1\right ) \cosh \left (\frac {2 c_1}{9}\right )-2 \left (x^2+1\right ) \sinh \left (\frac {2 c_1}{9}\right )+2 i}}}\\ y(x)&\to -x^2+\frac {1}{\frac {1}{x^2+1}+\frac {1+i}{\left (x^2+1\right ) \sqrt {-2 \left (x^2+1\right ) \cosh \left (\frac {2 c_1}{9}\right )-2 \left (x^2+1\right ) \sinh \left (\frac {2 c_1}{9}\right )+2 i}}}\\ y(x)&\to -x^2+\frac {1}{\frac {1}{x^2+1}-\frac {1+i}{\sqrt {2} \left (x^2+1\right ) \sqrt {\left (x^2+1\right ) \cosh \left (\frac {2 c_1}{9}\right )+\left (x^2+1\right ) \sinh \left (\frac {2 c_1}{9}\right )+i}}}\\ y(x)&\to -x^2+\frac {1}{\frac {1}{x^2+1}+\frac {1+i}{\sqrt {2} \left (x^2+1\right ) \sqrt {\left (x^2+1\right ) \cosh \left (\frac {2 c_1}{9}\right )+\left (x^2+1\right ) \sinh \left (\frac {2 c_1}{9}\right )+i}}}\\ y(x)&\to 1\\ y(x)&\to \frac {1}{2} \left (1-x^2\right ) \end{align*}

✓ Sympy. Time used: 0.794 (sec). Leaf size: 51

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

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