75.9.6 problem 225

Internal problem ID [16772]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 8.3. The Lagrange and Clairaut equations. Exercises page 72
Problem number : 225
Date solved : Monday, March 31, 2025 at 03:16:50 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _Clairaut]

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

✓ Maple. Time used: 0.042 (sec). Leaf size: 74

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

\begin{align*} y &= \frac {3 \,2^{{1}/{3}} \left (a \,x^{2}\right )^{{1}/{3}}}{2} \\ y &= -\frac {3 \,2^{{1}/{3}} \left (a \,x^{2}\right )^{{1}/{3}} \left (1+i \sqrt {3}\right )}{4} \\ y &= \frac {3 \,2^{{1}/{3}} \left (a \,x^{2}\right )^{{1}/{3}} \left (-1+i \sqrt {3}\right )}{4} \\ y &= \frac {c_1^{3} x +a}{c_1^{2}} \\ \end{align*}

✓ Mathematica. Time used: 0.016 (sec). Leaf size: 89

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

\begin{align*} y(x)\to \frac {a}{c_1{}^2}+c_1 x \\ y(x)\to \frac {3 \sqrt [3]{a} x^{2/3}}{2^{2/3}} \\ y(x)\to -\frac {3 \sqrt [3]{-1} \sqrt [3]{a} x^{2/3}}{2^{2/3}} \\ y(x)\to \frac {3 (-1)^{2/3} \sqrt [3]{a} x^{2/3}}{2^{2/3}} \\ \end{align*}

✗ Sympy

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

Timed Out