29.30.13 problem 872
Internal
problem
ID
[5453]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
30
Problem
number
:
872
Date
solved
:
Sunday, March 30, 2025 at 08:14:00 AM
CAS
classification
:
[[_homogeneous, `class A`], _rational, _dAlembert]
\begin{align*} x {y^{\prime }}^{2}+a y y^{\prime }+b x&=0 \end{align*}
✓ Maple. Time used: 0.052 (sec). Leaf size: 205
ode:=x*diff(y(x),x)^2+a*y(x)*diff(y(x),x)+b*x = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
\frac {c_1 2^{\frac {a +2}{2 a +2}} \left (y a +\sqrt {a^{2} y^{2}-4 x^{2} b}\right ) {\left (\frac {a \left (y \left (a +1\right ) \sqrt {a^{2} y^{2}-4 x^{2} b}+\left (a^{2}+a \right ) y^{2}-2 x^{2} b \right )}{x^{2}}\right )}^{\frac {-a -2}{2 a +2}}+x^{2}}{x} &= 0 \\
\frac {-c_1 \left (y a -\sqrt {a^{2} y^{2}-4 x^{2} b}\right ) {\left (\frac {\left (-y \left (a +1\right ) \sqrt {a^{2} y^{2}-4 x^{2} b}+\left (a^{2}+a \right ) y^{2}-2 x^{2} b \right ) a}{2 x^{2}}\right )}^{\frac {-a -2}{2 a +2}}+x^{2}}{x} &= 0 \\
\end{align*}
✓ Mathematica. Time used: 2.42 (sec). Leaf size: 423
ode=x (D[y[x],x])^2+a y[x] D[y[x],x]+b x==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
\text {Solve}\left [-\frac {i \left (2 \log \left (-i \sqrt {4 b-\frac {a^2 y(x)^2}{x^2}}+\frac {a y(x)}{x}+2 i \sqrt {b}\right )+2 (a+1) \log \left (i \sqrt {4 b-\frac {a^2 y(x)^2}{x^2}}+\frac {a y(x)}{x}-2 i \sqrt {b}\right )-(a+2) \log \left (\frac {i (a+2) y(x) \sqrt {4 b-\frac {a^2 y(x)^2}{x^2}}}{x}+2 \sqrt {b} \left (\sqrt {4 b-\frac {a^2 y(x)^2}{x^2}}-\frac {i (a+2) y(x)}{x}\right )+\frac {a^2 y(x)^2}{x^2}-4 b\right )\right )}{4 (a+1)}&=c_1-\frac {1}{2} i \log (x),y(x)\right ] \\
\text {Solve}\left [\frac {i \left (2 (a+1) \log \left (-i \sqrt {4 b-\frac {a^2 y(x)^2}{x^2}}+\frac {a y(x)}{x}+2 i \sqrt {b}\right )+2 \log \left (i \sqrt {4 b-\frac {a^2 y(x)^2}{x^2}}+\frac {a y(x)}{x}-2 i \sqrt {b}\right )-(a+2) \log \left (-\frac {i (a+2) y(x) \sqrt {4 b-\frac {a^2 y(x)^2}{x^2}}}{x}+2 \sqrt {b} \left (\sqrt {4 b-\frac {a^2 y(x)^2}{x^2}}+\frac {i (a+2) y(x)}{x}\right )+\frac {a^2 y(x)^2}{x^2}-4 b\right )\right )}{4 (a+1)}&=\frac {1}{2} i \log (x)+c_1,y(x)\right ] \\
\end{align*}
✓ Sympy. Time used: 26.357 (sec). Leaf size: 141
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 + x*Derivative(y(x), x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = C_{1} e^{- a \int \limits ^{\frac {x}{y{\left (x \right )}}} \frac {1}{u_{1} \left (a + \sqrt {- 4 u_{1}^{2} b + a^{2}} + 2\right )}\, du_{1} - \int \limits ^{\frac {x}{y{\left (x \right )}}} \frac {\sqrt {- 4 u_{1}^{2} b + a^{2}}}{u_{1} \left (a + \sqrt {- 4 u_{1}^{2} b + a^{2}} + 2\right )}\, du_{1}}, \ y{\left (x \right )} = C_{1} e^{- a \int \limits ^{\frac {x}{y{\left (x \right )}}} \frac {1}{u_{1} \left (a - \sqrt {- 4 u_{1}^{2} b + a^{2}} + 2\right )}\, du_{1} + \int \limits ^{\frac {x}{y{\left (x \right )}}} \frac {\sqrt {- 4 u_{1}^{2} b + a^{2}}}{u_{1} \left (a - \sqrt {- 4 u_{1}^{2} b + a^{2}} + 2\right )}\, du_{1}}\right ]
\]