66.1.27 problem Problem 39

Internal problem ID [13803]
Book : Differential equations and the calculus of variations by L. ElSGOLTS. MIR PUBLISHERS, MOSCOW, Third printing 1977.
Section : Chapter 1, First-Order Differential Equations. Problems page 88
Problem number : Problem 39
Date solved : Monday, March 31, 2025 at 08:13:01 AM
CAS classification : [[_1st_order, _with_linear_symmetries], _dAlembert]

\begin{align*} y&=5 x y^{\prime }-{y^{\prime }}^{2} \end{align*}

✓ Maple. Time used: 0.030 (sec). Leaf size: 93

ode:=y(x) = 5*x*diff(y(x),x)-diff(y(x),x)^2; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} -\frac {4 \sqrt {2}\, c_1}{\left (10 x -2 \sqrt {25 x^{2}-4 y}\right )^{{5}/{4}}}+\frac {4 x}{9}+\frac {\sqrt {25 x^{2}-4 y}}{9} &= 0 \\ -\frac {4 \sqrt {2}\, c_1}{\left (10 x +2 \sqrt {25 x^{2}-4 y}\right )^{{5}/{4}}}+\frac {4 x}{9}-\frac {\sqrt {25 x^{2}-4 y}}{9} &= 0 \\ \end{align*}

✓ Mathematica. Time used: 41.801 (sec). Leaf size: 2238

ode=y[x]==5*x*D[y[x],x]-D[y[x],x]^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Too large to display

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-5*x*Derivative(y(x), x) + y(x) + Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -5*x/2 - sqrt(25*x**2 - 4*y(x))/2 + Derivative(y(x), x) cannot be solved by the factorable group method