56.14.4 problem Ex 4

Internal problem ID [14166]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter IV, differential equations of the first order and higher degree than the first. Article 25. Equations solvable for \(y\). Page 52
Problem number : Ex 4
Date solved : Thursday, October 02, 2025 at 09:17:58 AM
CAS classification : [[_homogeneous, `class C`], _Riccati]

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

✓ Maple. Time used: 0.011 (sec). Leaf size: 29

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

\[ y = \frac {c_1 \left (-1+x \right ) {\mathrm e}^{2 x}-x -1}{-1+{\mathrm e}^{2 x} c_1} \]

✓ Mathematica. Time used: 0.121 (sec). Leaf size: 29

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

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

✓ Sympy. Time used: 0.143 (sec). Leaf size: 26

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

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