40.6.1 problem 10

Internal problem ID [6681]
Book : Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres. McGraw Hill 1952
Section : Chapter 10. Singular solutions, Extraneous loci. Supplemetary problems. Page 74
Problem number : 10
Date solved : Wednesday, March 05, 2025 at 02:35:15 AM
CAS classification : [[_1st_order, _with_linear_symmetries], _Clairaut]

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

✓ Maple. Time used: 0.015 (sec). Leaf size: 19

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

\begin{align*} y &= \frac {x^{2}}{8} \\ y &= c_1 \left (-2 c_1 +x \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 25

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

\begin{align*} y(x)\to c_1 (x-2 c_1) \\ y(x)\to \frac {x^2}{8} \\ \end{align*}

✓ Sympy. Time used: 1.827 (sec). Leaf size: 14

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

\[ y{\left (x \right )} = \frac {x^{2}}{8} - \frac {\left (C_{1} + x\right )^{2}}{8} \]