69.6.36 problem 169

Internal problem ID [18079]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 6. Linear equations of the first order. The Bernoulli equation. Exercises page 54
Problem number : 169
Date solved : Thursday, October 02, 2025 at 02:37:33 PM
CAS classification : [`y=_G(x,y')`]

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

✓ Maple. Time used: 0.020 (sec). Leaf size: 42

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

\begin{align*} y &= \sqrt {2}\, \sqrt {\ln \left (-{\mathrm e}^{-x} c_1 +x -2\right )} \\ y &= -\sqrt {2}\, \sqrt {\ln \left (-{\mathrm e}^{-x} c_1 +x -2\right )} \\ \end{align*}

✓ Mathematica. Time used: 6.285 (sec). Leaf size: 60

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

\begin{align*} y(x)&\to -\sqrt {2} \sqrt {-x+\log \left (e^x (x-2)+c_1\right )}\\ y(x)&\to \sqrt {2} \sqrt {-x+\log \left (e^x (x-2)+c_1\right )} \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE -(x - exp(y(x)**2/2) - 1)*exp(-y(x)**2/2)/y(x) + Derivative(y(x), x) cannot be solved by the factorable group method