40.5.15 problem 31
Internal
problem
ID
[6680]
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
:
Sunday, March 30, 2025 at 11:18:01 AM
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: 60.269 (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: 33.741 (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)
\[
\left [ \log {\left (x \right )} = C_{1} + \log {\left (\frac {\sqrt {2} \left (\sqrt {1 - \frac {12 y^{2}{\left (x \right )}}{x^{2}}} - 1\right )^{\frac {3}{4}}}{2 \left (\sqrt {1 - \frac {12 y^{2}{\left (x \right )}}{x^{2}}} - 5\right )^{\frac {3}{4}} \sqrt {\sqrt {1 - \frac {12 y^{2}{\left (x \right )}}{x^{2}}} - 1 + \frac {2 y^{2}{\left (x \right )}}{x^{2}}}} \right )}, \ y{\left (x \right )} = 0\right ]
\]