85.4.5 problem 1 (e)

Internal problem ID [22446]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter 1. Differential equations in general. Section 1.3. A Exercises at page 21
Problem number : 1 (e)
Date solved : Thursday, October 02, 2025 at 08:39:37 PM
CAS classification : [[_2nd_order, _exact, _nonlinear], _Liouville, [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

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

With initial conditions

\begin{align*} y \left (3\right )&=1 \\ y^{\prime }\left (3\right )&=2 \\ \end{align*}

✓ Maple. Time used: 0.039 (sec). Leaf size: 19

ode:=x*diff(y(x),x)^2+2*y(x)*diff(y(x),x)+x*y(x)*diff(diff(y(x),x),x) = 0; 
ic:=[y(3) = 1, D(y)(3) = 2]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {\sqrt {13 x^{2}-36 x}}{x} \]

✓ Mathematica. Time used: 0.212 (sec). Leaf size: 23

ode=x*D[y[x],{x,1}]^2+2*y[x]*D[y[x],{x,1}]+x*y[x]*D[y[x],{x,2}]==0; 
ic={y[3]==1,Derivative[1][y][3] ==2}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {i \sqrt {36-13 x}}{\sqrt {x}} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*y(x)*Derivative(y(x), (x, 2)) + x*Derivative(y(x), x)**2 + 2*y(x)*Derivative(y(x), x),0) 
ics = {y(3): 1, Subs(Derivative(y(x), x), x, 3): 2} 
dsolve(ode,func=y(x),ics=ics)
 

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