23.2.45 problem 46

Internal problem ID [5400]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 2. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF SECOND OR HIGHER DEGREE, page 278
Problem number : 46
Date solved : Tuesday, September 30, 2025 at 12:39:51 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _Clairaut]

\begin{align*} {y^{\prime }}^{2}-2 x y^{\prime }+2 y&=0 \end{align*}

✓ Maple. Time used: 0.030 (sec). Leaf size: 20

ode:=diff(y(x),x)^2-2*x*diff(y(x),x)+2*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {x^{2}}{2} \\ y &= -\frac {c_1 \left (c_1 -2 x \right )}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 29

ode=(D[y[x],x])^2-2*x*D[y[x],x]+2*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.924 (sec). Leaf size: 14

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

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