83.10.13 problem 13

Internal problem ID [21982]
Book : Differential Equations By Kaj L. Nielsen. Second edition 1966. Barnes and nobel. 66-28306
Section : Chapter IV. First order differential equations of higher degree. Ex. XI at page 69
Problem number : 13
Date solved : Thursday, October 02, 2025 at 08:21:23 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _Clairaut]

\begin{align*} {y^{\prime }}^{2}+y&=x y^{\prime }+1 \end{align*}

✓ Maple. Time used: 0.012 (sec). Leaf size: 24

ode:=y(x)+diff(y(x),x)^2 = x*diff(y(x),x)+1; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 30

ode=y[x]+D[y[x],x]^2==x*D[y[x],x]+1; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 1.017 (sec). Leaf size: 14

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*Derivative(y(x), x) + y(x) + Derivative(y(x), x)**2 - 1,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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