29.29.33 problem 855
Internal
problem
ID
[5438]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
29
Problem
number
:
855
Date
solved
:
Tuesday, March 04, 2025 at 09:39:20 PM
CAS
classification
:
[[_homogeneous, `class A`], _rational, _dAlembert]
\begin{align*} x {y^{\prime }}^{2}-y y^{\prime }+a x&=0 \end{align*}
✓ Maple. Time used: 0.066 (sec). Leaf size: 50
ode:=x*diff(y(x),x)^2-y(x)*diff(y(x),x)+a*x = 0;
dsolve(ode,y(x), singsol=all);
\[
y \left (x \right ) = \left (-\operatorname {LambertW}\left (-\frac {x^{2}}{c_{1}^{2} a}\right )+1\right ) a c_{1} \sqrt {-\frac {x^{2}}{c_{1}^{2} a \operatorname {LambertW}\left (-\frac {x^{2}}{c_{1}^{2} a}\right )}}
\]
✓ Mathematica. Time used: 4.248 (sec). Leaf size: 239
ode=x (D[y[x],x])^2-y[x] D[y[x],x]+a x==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
\text {Solve}\left [\frac {i a}{\left (\sqrt {4 a-\frac {y(x)^2}{x^2}}-\frac {i y(x)}{x}\right )^2}+\frac {y(x)}{x \left (\sqrt {4 a-\frac {y(x)^2}{x^2}}-\frac {i y(x)}{x}\right )}+\frac {1}{2} i \log \left (\sqrt {4 a-\frac {y(x)^2}{x^2}}-\frac {i y(x)}{x}\right )&=c_1-\frac {1}{2} i \log (x),y(x)\right ] \\
\text {Solve}\left [-\frac {i a}{\left (\sqrt {4 a-\frac {y(x)^2}{x^2}}+\frac {i y(x)}{x}\right )^2}+\frac {y(x)}{x \left (\sqrt {4 a-\frac {y(x)^2}{x^2}}+\frac {i y(x)}{x}\right )}-\frac {1}{2} i \log \left (\sqrt {4 a-\frac {y(x)^2}{x^2}}+\frac {i y(x)}{x}\right )&=\frac {1}{2} i \log (x)+c_1,y(x)\right ] \\
\end{align*}
✓ Sympy. Time used: 37.188 (sec). Leaf size: 275
from sympy import *
x = symbols("x")
a = symbols("a")
y = Function("y")
ode = Eq(a*x + x*Derivative(y(x), x)**2 - y(x)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ \log {\left (x \right )} = C_{1} - \log {\left (\left (\sqrt {- 4 a + \frac {y^{2}{\left (x \right )}}{x^{2}}} + \frac {y{\left (x \right )}}{x}\right )^{\frac {\sqrt {- 4 a + \frac {y^{2}{\left (x \right )}}{x^{2}}}}{\sqrt {- 4 a + \frac {y^{2}{\left (x \right )}}{x^{2}}} + \frac {y{\left (x \right )}}{x}}} \left (\sqrt {- 4 a + \frac {y^{2}{\left (x \right )}}{x^{2}}} + \frac {y{\left (x \right )}}{x}\right )^{\frac {y{\left (x \right )}}{x \left (\sqrt {- 4 a + \frac {y^{2}{\left (x \right )}}{x^{2}}} + \frac {y{\left (x \right )}}{x}\right )}} \right )} - \frac {y{\left (x \right )}}{x \left (\sqrt {- 4 a + \frac {y^{2}{\left (x \right )}}{x^{2}}} + \frac {y{\left (x \right )}}{x}\right )}, \ \log {\left (x \right )} = C_{1} + \frac {\sqrt {- 4 a + \frac {y^{2}{\left (x \right )}}{x^{2}}}}{\sqrt {- 4 a + \frac {y^{2}{\left (x \right )}}{x^{2}}} - \frac {y{\left (x \right )}}{x}} + \log {\left (\left (\sqrt {- 4 a + \frac {y^{2}{\left (x \right )}}{x^{2}}} - \frac {y{\left (x \right )}}{x}\right )^{- \frac {\sqrt {- 4 a + \frac {y^{2}{\left (x \right )}}{x^{2}}}}{\sqrt {- 4 a + \frac {y^{2}{\left (x \right )}}{x^{2}}} - \frac {y{\left (x \right )}}{x}}} \left (\sqrt {- 4 a + \frac {y^{2}{\left (x \right )}}{x^{2}}} - \frac {y{\left (x \right )}}{x}\right )^{\frac {y{\left (x \right )}}{x \left (\sqrt {- 4 a + \frac {y^{2}{\left (x \right )}}{x^{2}}} - \frac {y{\left (x \right )}}{x}\right )}} \right )}\right ]
\]