40.6.2 problem 11

Internal problem ID [6682]
Book : Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres. McGraw Hill 1952
Section : Chapter 10. Singular solutions, Extraneous loci. Supplemetary problems. Page 74
Problem number : 11
Date solved : Wednesday, March 05, 2025 at 02:35:16 AM
CAS classification : [[_1st_order, _with_linear_symmetries], _rational]

\begin{align*} y^{2} {y^{\prime }}^{2}+3 x y^{\prime }-y&=0 \end{align*}

✓ Maple. Time used: 0.054 (sec). Leaf size: 123

ode:=y(x)^2*diff(y(x),x)^2+3*x*diff(y(x),x)-y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {18^{{1}/{3}} \left (-x^{2}\right )^{{1}/{3}}}{2} \\ y &= -\frac {2^{{1}/{3}} \left (-x^{2}\right )^{{1}/{3}} \left (3 i 3^{{1}/{6}}+3^{{2}/{3}}\right )}{4} \\ y &= \frac {2^{{1}/{3}} \left (-x^{2}\right )^{{1}/{3}} \left (-3^{{2}/{3}}+3 i 3^{{1}/{6}}\right )}{4} \\ y &= 0 \\ y &= \operatorname {RootOf}\left (-2 \ln \left (x \right )-3 \left (\int _{}^{\textit {\_Z}}\frac {4 \textit {\_a}^{3}+3 \sqrt {4 \textit {\_a}^{3}+9}+9}{\textit {\_a} \left (4 \textit {\_a}^{3}+9\right )}d \textit {\_a} \right )+2 c_1 \right ) x^{{2}/{3}} \\ \end{align*}

✓ Mathematica. Time used: 0.542 (sec). Leaf size: 204

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

\begin{align*} \text {Solve}\left [\frac {3}{2} \log (y(x))-\frac {2 \sqrt {\frac {9 x^2}{4 y(x)^3}+1} \text {arcsinh}\left (\frac {3}{2} x \sqrt {\frac {1}{y(x)^3}}\right )}{\sqrt {\frac {1}{y(x)^3}} \sqrt {9 x^2+4 y(x)^3}}&=c_1,y(x)\right ] \\ \text {Solve}\left [\frac {2 \sqrt {\frac {9 x^2}{4 y(x)^3}+1} \text {arcsinh}\left (\frac {3}{2} x \sqrt {\frac {1}{y(x)^3}}\right )}{\sqrt {\frac {1}{y(x)^3}} \sqrt {9 x^2+4 y(x)^3}}+\frac {3}{2} \log (y(x))&=c_1,y(x)\right ] \\ y(x)\to -\left (-\frac {3}{2}\right )^{2/3} x^{2/3} \\ y(x)\to -\left (\frac {3}{2}\right )^{2/3} x^{2/3} \\ y(x)\to \frac {\sqrt [3]{-1} 3^{2/3} x^{2/3}}{2^{2/3}} \\ \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE -(-3*x + sqrt(9*x**2 + 4*y(x)**3))/(2*y(x)**2) + Derivative(y(x), x) cannot be solved by the factorable group method