60.1.452 problem 465
Internal
problem
ID
[10466]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
465
Date
solved
:
Sunday, March 30, 2025 at 04:50:46 PM
CAS
classification
:
[[_homogeneous, `class A`], _rational, _dAlembert]
\begin{align*} y {y^{\prime }}^{2}+2 x y^{\prime }-9 y&=0 \end{align*}
✓ Maple. Time used: 0.168 (sec). Leaf size: 92
ode:=y(x)*diff(y(x),x)^2+2*x*diff(y(x),x)-9*y(x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= 0 \\
y &= \operatorname {RootOf}\left (-\ln \left (x \right )-\int _{}^{\textit {\_Z}}\frac {\textit {\_a}^{2}+\sqrt {9 \textit {\_a}^{2}+1}+1}{\textit {\_a} \left (\textit {\_a}^{2}-7\right )}d \textit {\_a} +c_1 \right ) x \\
y &= \operatorname {RootOf}\left (-\ln \left (x \right )+\int _{}^{\textit {\_Z}}-\frac {\textit {\_a}^{2}-\sqrt {9 \textit {\_a}^{2}+1}+1}{\textit {\_a} \left (\textit {\_a}^{2}-7\right )}d \textit {\_a} +c_1 \right ) x \\
\end{align*}
✓ Mathematica. Time used: 0.128 (sec). Leaf size: 112
ode=-9*y[x] + 2*x*D[y[x],x] + y[x]*D[y[x],x]^2==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
\text {Solve}\left [\int \frac {y(x)}{x \left (\frac {y(x)^2}{x^2}-\sqrt {\frac {9 y(x)^2}{x^2}+1}+1\right )}d\frac {y(x)}{x}&=-\log (x)+c_1,y(x)\right ] \\
\text {Solve}\left [\int \frac {y(x)}{x \left (\frac {y(x)^2}{x^2}+\sqrt {\frac {9 y(x)^2}{x^2}+1}+1\right )}d\frac {y(x)}{x}&=-\log (x)+c_1,y(x)\right ] \\
y(x)\to 0 \\
\end{align*}
✓ Sympy. Time used: 22.769 (sec). Leaf size: 85
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(2*x*Derivative(y(x), x) + y(x)*Derivative(y(x), x)**2 - 9*y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = 0, \ \log {\left (x \right )} = C_{1} + \log {\left (\frac {\sqrt {2} \left (\sqrt {1 + \frac {9 y^{2}{\left (x \right )}}{x^{2}}} - 1\right )^{\frac {9}{14}}}{2 \left (\sqrt {1 + \frac {9 y^{2}{\left (x \right )}}{x^{2}}} - 8\right )^{\frac {9}{14}} \sqrt {- \sqrt {1 + \frac {9 y^{2}{\left (x \right )}}{x^{2}}} + 1 + \frac {y^{2}{\left (x \right )}}{x^{2}}}} \right )}\right ]
\]