62.7.2 problem Ex 2

Internal problem ID [12754]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter 2, differential equations of the first order and the first degree. Article 14. Equations reducible to linear equations (Bernoulli). Page 21
Problem number : Ex 2
Date solved : Monday, March 31, 2025 at 07:02:08 AM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 33

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

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

✓ Mathematica. Time used: 1.933 (sec). Leaf size: 57

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

\begin{align*} y(x)\to -\sqrt {1+e^{-x^2+2 c_1}} \\ y(x)\to \sqrt {1+e^{-x^2+2 c_1}} \\ y(x)\to -1 \\ y(x)\to 1 \\ \end{align*}

✓ Sympy. Time used: 0.566 (sec). Leaf size: 29

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

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