25.6.12 problem 12

Internal problem ID [6896]
Book : Differential Equations, By George Boole F.R.S. 1865
Section : Chapter 7
Problem number : 12
Date solved : Tuesday, September 30, 2025 at 04:00:26 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _Clairaut]

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

✓ Maple. Time used: 0.033 (sec). Leaf size: 22

ode:=y(x) = x*diff(y(x),x)+diff(y(x),x)-diff(y(x),x)^2; 
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: 0.005 (sec). Leaf size: 28

ode=y[x]==x*D[y[x],x]+D[y[x],x]-(D[y[x],x])^2; 
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: 1.349 (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)
 

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