29.30.12 problem 871
Internal
problem
ID
[5452]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
30
Problem
number
:
871
Date
solved
:
Sunday, March 30, 2025 at 08:13:58 AM
CAS
classification
:
[[_homogeneous, `class G`], _rational, _dAlembert]
\begin{align*} x {y^{\prime }}^{2}-a y y^{\prime }+b&=0 \end{align*}
✓ Maple. Time used: 0.041 (sec). Leaf size: 339
ode:=x*diff(y(x),x)^2-a*y(x)*diff(y(x),x)+b = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
\frac {4 c_1 \left (\left (a -\frac {1}{2}\right )^{2} a y 2^{-\frac {1}{a -1}} \sqrt {a^{2} y^{2}-4 x b}-\frac {b x 2^{\frac {a -2}{a -1}}}{4}+\left (\left (a -\frac {1}{2}\right )^{2} a y^{2}-2 x b \left (a -1\right )\right ) a 2^{-\frac {1}{a -1}}\right ) {\left (\frac {y a +\sqrt {a^{2} y^{2}-4 x b}}{x}\right )}^{\frac {1}{a -1}}+4 a \left (y \left (a -\frac {1}{2}\right ) \sqrt {a^{2} y^{2}-4 x b}+\left (a^{2}-\frac {1}{2} a \right ) y^{2}-2 x b \right ) x}{\left (2 a -1\right ) \left (y a +\sqrt {a^{2} y^{2}-4 x b}\right )^{2}} &= 0 \\
\frac {-4 \left (a -\frac {1}{2}\right )^{2} \left (a^{2} y^{2}-a y \sqrt {a^{2} y^{2}-4 x b}-2 x b \right ) c_1 {\left (-\frac {-y a +\sqrt {a^{2} y^{2}-4 x b}}{2 x}\right )}^{\frac {1}{a -1}}+4 a \left (-y \left (a -\frac {1}{2}\right ) \sqrt {a^{2} y^{2}-4 x b}+\left (a^{2}-\frac {1}{2} a \right ) y^{2}-2 x b \right ) x}{\left (2 a -1\right ) \left (y a -\sqrt {a^{2} y^{2}-4 x b}\right )^{2}} &= 0 \\
\end{align*}
✓ Mathematica. Time used: 0.831 (sec). Leaf size: 143
ode=x (D[y[x],x])^2-a y[x] D[y[x],x]+b==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
\text {Solve}\left [\frac {2 \left ((a-1) \log \left (\sqrt {a^2 y(x)^2-4 b x}+(a-1) y(x)\right )+a \log \left (\sqrt {a^2 y(x)^2-4 b x}-a y(x)\right )\right )}{2 a-1}&=c_1,y(x)\right ] \\
\text {Solve}\left [\frac {2 \left ((a-1) \log \left (\sqrt {a^2 y(x)^2-4 b x}-a y(x)+y(x)\right )+a \log \left (\sqrt {a^2 y(x)^2-4 b x}+a y(x)\right )\right )}{2 a-1}&=c_1,y(x)\right ] \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
y = Function("y")
ode = Eq(-a*y(x)*Derivative(y(x), x) + b + x*Derivative(y(x), x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out