15.19.24 problem 24

Internal problem ID [3308]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 37, page 171
Problem number : 24
Date solved : Sunday, March 30, 2025 at 01:34:29 AM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.031 (sec). Leaf size: 118

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

\begin{align*} x \left (1-\frac {c_1}{\left (\frac {-2 x +\sqrt {x \left (x +y\right )}}{x}\right )^{{1}/{3}} \left (\frac {x +\sqrt {x \left (x +y\right )}}{x}\right )^{{2}/{3}} \left (-2 x +\sqrt {x \left (x +y\right )}\right )}\right ) &= 0 \\ x \left (1+\frac {c_1}{\left (\frac {-2 x -\sqrt {x \left (x +y\right )}}{x}\right )^{{1}/{3}} \left (\frac {x -\sqrt {x \left (x +y\right )}}{x}\right )^{{2}/{3}} \left (2 x +\sqrt {x \left (x +y\right )}\right )}\right ) &= 0 \\ \end{align*}

✓ Mathematica. Time used: 60.099 (sec). Leaf size: 1178

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

\begin{align*} \text {Solution too large to show}\end{align*}

✓ Sympy. Time used: 19.027 (sec). Leaf size: 56

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

\[ \left [ \log {\left (x \right )} = C_{1} + \log {\left (\frac {\sqrt [3]{\sqrt {1 + \frac {y{\left (x \right )}}{x}} - 1}}{\sqrt [3]{\sqrt {1 + \frac {y{\left (x \right )}}{x}} + 2} \left (\sqrt {1 + \frac {y{\left (x \right )}}{x}} - 1 + \frac {y{\left (x \right )}}{x}\right )} \right )}, \ y{\left (x \right )} = 0\right ] \]