69.1.2 problem 2

Internal problem ID [14084]
Book : DIFFERENTIAL and INTEGRAL CALCULUS. VOL I. by N. PISKUNOV. MIR PUBLISHERS, Moscow 1969.
Section : Chapter 8. Differential equations. Exercises page 595
Problem number : 2
Date solved : Monday, March 31, 2025 at 12:03:03 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _Clairaut]

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

✓ Maple. Time used: 0.029 (sec). Leaf size: 22

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

\begin{align*} y &= \frac {\left (x +1\right )^{2}}{4} \\ y &= c_1 \left (-c_1 +x +1\right ) \\ \end{align*}

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 28

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

\begin{align*} y(x)\to c_1 (x+1-c_1) \\ y(x)\to \frac {1}{4} (x+1)^2 \\ \end{align*}

✓ Sympy. Time used: 2.067 (sec). Leaf size: 20

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

\[ y{\left (x \right )} = \frac {x^{2}}{4} + \frac {x}{2} - \frac {\left (C_{1} + x\right )^{2}}{4} + \frac {1}{4} \]