71.1.32 problem 49 (page 56)

Internal problem ID [19208]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 49 (page 56)
Date solved : Thursday, October 02, 2025 at 03:43:40 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _rational, [_Abel, `2nd type`, `class B`]]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 115

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

\begin{align*} y &= \frac {c_1 \,a^{2} b^{2} x +x +\sqrt {\left (a^{2}+b^{2}\right ) \left (-1+\left (a^{4}+a^{2} x^{2}+b^{2} x^{2}\right ) c_1 \right )}}{c_1 \,a^{4}-1} \\ y &= \frac {c_1 \,a^{2} b^{2} x +x -\sqrt {\left (a^{2}+b^{2}\right ) \left (-1+\left (a^{4}+a^{2} x^{2}+b^{2} x^{2}\right ) c_1 \right )}}{c_1 \,a^{4}-1} \\ \end{align*}

✓ Mathematica. Time used: 4.461 (sec). Leaf size: 1530

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

\begin{align*} \text {Solution too large to show}\end{align*}

✓ Sympy. Time used: 53.947 (sec). Leaf size: 54

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

\[ C_{1} + \frac {a^{2} \log {\left (- a^{2} y{\left (x \right )} + b^{2} x \right )}}{a^{2} + b^{2}} - \frac {a^{2} \log {\left (a^{2} + b^{2} + \left (x + y{\left (x \right )}\right )^{2} \right )}}{2 \left (a^{2} + b^{2}\right )} = 0 \]