9.19.19 problem 19
Internal
problem
ID
[3303]
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
:
19
Date
solved
:
Tuesday, September 30, 2025 at 06:33:02 AM
CAS
classification
:
[[_homogeneous, `class A`], _rational, _dAlembert]
\begin{align*} {y^{\prime }}^{2} y&=3 y^{\prime } x +y \end{align*}
✓ Maple. Time used: 0.885 (sec). Leaf size: 273
ode:=y(x)*diff(y(x),x)^2 = 3*x*diff(y(x),x)+y(x);
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= 0 \\
\ln \left (x \right )-\frac {3 \,\operatorname {arctanh}\left (\frac {3}{\sqrt {\frac {9 x^{2}+4 y^{2}}{x^{2}}}}\right )}{8}+\frac {5 \,\operatorname {arctanh}\left (\frac {9 x +8 y}{5 x \sqrt {\frac {9 x^{2}+4 y^{2}}{x^{2}}}}\right )}{16}-\frac {5 \,\operatorname {arctanh}\left (\frac {-9 x +8 y}{5 x \sqrt {\frac {9 x^{2}+4 y^{2}}{x^{2}}}}\right )}{16}+\frac {5 \ln \left (\frac {y+2 x}{x}\right )}{16}+\frac {5 \ln \left (\frac {y-2 x}{x}\right )}{16}+\frac {3 \ln \left (\frac {y}{x}\right )}{8}-c_1 &= 0 \\
\ln \left (x \right )+\frac {3 \,\operatorname {arctanh}\left (\frac {3}{\sqrt {\frac {9 x^{2}+4 y^{2}}{x^{2}}}}\right )}{8}-\frac {5 \,\operatorname {arctanh}\left (\frac {9 x +8 y}{5 x \sqrt {\frac {9 x^{2}+4 y^{2}}{x^{2}}}}\right )}{16}+\frac {5 \,\operatorname {arctanh}\left (\frac {-9 x +8 y}{5 x \sqrt {\frac {9 x^{2}+4 y^{2}}{x^{2}}}}\right )}{16}+\frac {5 \ln \left (\frac {y+2 x}{x}\right )}{16}+\frac {5 \ln \left (\frac {y-2 x}{x}\right )}{16}+\frac {3 \ln \left (\frac {y}{x}\right )}{8}-c_1 &= 0 \\
\end{align*}
✓ Mathematica. Time used: 71.44 (sec). Leaf size: 2113
ode=D[y[x],x]^2*y[x]==3*D[y[x],x]*x+y[x];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
Too large to display
✓ Sympy. Time used: 15.827 (sec). Leaf size: 87
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-3*x*Derivative(y(x), x) + y(x)*Derivative(y(x), x)**2 - y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ \log {\left (x \right )} = C_{1} + \log {\left (\frac {\sqrt {2} \sqrt [8]{\sqrt {9 + \frac {4 y^{2}{\left (x \right )}}{x^{2}}} - 3}}{2 \sqrt [8]{\sqrt {9 + \frac {4 y^{2}{\left (x \right )}}{x^{2}}} + 5} \sqrt {\sqrt {9 + \frac {4 y^{2}{\left (x \right )}}{x^{2}}} - 3 + \frac {2 y^{2}{\left (x \right )}}{x^{2}}}} \right )}, \ y{\left (x \right )} = 0\right ]
\]