34.5.15 problem 31

Internal problem ID [7967]
Book : Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres. McGraw Hill 1952
Section : Chapter 9. Equations of first order and higher degree. Supplemetary problems. Page 65
Problem number : 31
Date solved : Tuesday, September 30, 2025 at 05:12:28 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.434 (sec). Leaf size: 159

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

\begin{align*} y &= 0 \\ \ln \left (x \right )-\frac {\operatorname {arctanh}\left (\frac {1}{\sqrt {-\frac {-x^{2}+12 y^{2}}{x^{2}}}}\right )}{4}+\frac {5 \,\operatorname {arctanh}\left (\frac {\sqrt {\frac {x^{2}-12 y^{2}}{x^{2}}}}{5}\right )}{4}-\frac {\ln \left (\frac {y}{x}\right )}{4}+\frac {5 \ln \left (\frac {2 x^{2}+y^{2}}{x^{2}}\right )}{8}-c_1 &= 0 \\ \ln \left (x \right )+\frac {\operatorname {arctanh}\left (\frac {1}{\sqrt {-\frac {-x^{2}+12 y^{2}}{x^{2}}}}\right )}{4}-\frac {5 \,\operatorname {arctanh}\left (\frac {\sqrt {\frac {x^{2}-12 y^{2}}{x^{2}}}}{5}\right )}{4}-\frac {\ln \left (\frac {y}{x}\right )}{4}+\frac {5 \ln \left (\frac {2 x^{2}+y^{2}}{x^{2}}\right )}{8}-c_1 &= 0 \\ \end{align*}

✓ Mathematica. Time used: 60.171 (sec). Leaf size: 1131

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

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

✓ Sympy. Time used: 17.139 (sec). Leaf size: 87

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

\[ \left [ \log {\left (x \right )} = C_{1} + \log {\left (\frac {\sqrt {2} \left (\sqrt {1 - \frac {12 y^{2}{\left (x \right )}}{x^{2}}} - 1\right )^{\frac {3}{4}}}{2 \left (\sqrt {1 - \frac {12 y^{2}{\left (x \right )}}{x^{2}}} - 5\right )^{\frac {3}{4}} \sqrt {\sqrt {1 - \frac {12 y^{2}{\left (x \right )}}{x^{2}}} - 1 + \frac {2 y^{2}{\left (x \right )}}{x^{2}}}} \right )}, \ y{\left (x \right )} = 0\right ] \]