69.6.24 problem 157

Internal problem ID [18067]
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 : 157
Date solved : Thursday, October 02, 2025 at 02:36:51 PM
CAS classification : [_separable]

\begin{align*} y^{\prime }+2 y x&=2 x y^{2} \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 14

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

\[ y = \frac {1}{1+{\mathrm e}^{x^{2}} c_1} \]

✓ Mathematica. Time used: 0.145 (sec). Leaf size: 42

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

\begin{align*} y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{(K[1]-1) K[1]}dK[1]\&\right ]\left [x^2+c_1\right ]\\ y(x)&\to 0\\ y(x)&\to 1 \end{align*}

✓ Sympy. Time used: 0.244 (sec). Leaf size: 12

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

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