56.1.17 problem 17

Internal problem ID [8729]
Book : Own collection of miscellaneous problems
Section : section 1.0
Problem number : 17
Date solved : Wednesday, March 05, 2025 at 06:14:09 AM
CAS classification : [[_1st_order, _with_linear_symmetries], _Clairaut]

\begin{align*} \frac {{y^{\prime }}^{2}}{4}-x y^{\prime }+y&=0 \end{align*}

✓ Maple. Time used: 0.025 (sec). Leaf size: 18

ode:=1/4*diff(y(x),x)^2-x*diff(y(x),x)+y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 25

ode=(1/4)*(D[y[x],x])^2-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-\frac {c_1{}^2}{4} \\ y(x)\to x^2 \\ \end{align*}

✓ Sympy. Time used: 1.668 (sec). Leaf size: 10

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*Derivative(y(x), x) + y(x) + Derivative(y(x), x)**2/4,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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