67.7.7 problem 7

Internal problem ID [16452]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 8. Review exercises for part of part II. page 143
Problem number : 7
Date solved : Thursday, October 02, 2025 at 01:33:46 PM
CAS classification : [[_homogeneous, `class C`], _Riccati]

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

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

ode:=diff(y(x),x) = y(x)^2-2*x*y(x)+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: 0.084 (sec). Leaf size: 29

ode=D[y[x],x]==y[x]^2-2*x*y[x]+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: 0.145 (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)
 

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