29.32.16 problem 950
Internal
problem
ID
[5528]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
32
Problem
number
:
950
Date
solved
:
Tuesday, March 04, 2025 at 09:51:20 PM
CAS
classification
:
[[_homogeneous, `class A`], _rational, _dAlembert]
\begin{align*} y {y^{\prime }}^{2}-\left (x +y\right ) y^{\prime }+y&=0 \end{align*}
✓ Maple. Time used: 0.076 (sec). Leaf size: 176
ode:=y(x)*diff(y(x),x)^2-(x+y(x))*diff(y(x),x)+y(x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y \left (x \right ) &= x \\
y \left (x \right ) &= 0 \\
\frac {-x \sqrt {\frac {\left (3 y \left (x \right )+x \right ) \left (x -y \left (x \right )\right )}{x^{2}}}+2 y \left (x \right ) \ln \left (\frac {y \left (x \right )}{x}\right )+\left (-2 \,\operatorname {arctanh}\left (\frac {x +y \left (x \right )}{x \sqrt {\frac {\left (3 y \left (x \right )+x \right ) \left (x -y \left (x \right )\right )}{x^{2}}}}\right )-2 c_{1} +2 \ln \left (x \right )\right ) y \left (x \right )-x}{2 y \left (x \right )} &= 0 \\
\frac {x \sqrt {\frac {\left (3 y \left (x \right )+x \right ) \left (x -y \left (x \right )\right )}{x^{2}}}+2 y \left (x \right ) \ln \left (\frac {y \left (x \right )}{x}\right )+\left (2 \,\operatorname {arctanh}\left (\frac {x +y \left (x \right )}{x \sqrt {\frac {\left (3 y \left (x \right )+x \right ) \left (x -y \left (x \right )\right )}{x^{2}}}}\right )-2 c_{1} +2 \ln \left (x \right )\right ) y \left (x \right )-x}{2 y \left (x \right )} &= 0 \\
\end{align*}
✓ Mathematica. Time used: 2.348 (sec). Leaf size: 192
ode=y[x] (D[y[x],x])^2-(x+y[x])D[y[x],x]+y[x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
\text {Solve}\left [\log \left (\sqrt {\frac {y(x)}{x}-1}+i \sqrt {\frac {3 y(x)}{x}+1}\right )-\frac {i \sqrt {\frac {3 y(x)}{x}+1}}{\sqrt {\frac {y(x)}{x}-1}+i \sqrt {\frac {3 y(x)}{x}+1}}&=-\frac {\log (x)}{2}+c_1,y(x)\right ] \\
\text {Solve}\left [\log \left (\sqrt {\frac {y(x)}{x}-1}-i \sqrt {\frac {3 y(x)}{x}+1}\right )-\frac {\sqrt {\frac {3 y(x)}{x}+1}}{\sqrt {\frac {3 y(x)}{x}+1}+i \sqrt {\frac {y(x)}{x}-1}}&=-\frac {\log (x)}{2}+c_1,y(x)\right ] \\
y(x)\to 0 \\
\end{align*}
✓ Sympy. Time used: 45.006 (sec). Leaf size: 196
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq((-x - y(x))*Derivative(y(x), x) + y(x)*Derivative(y(x), x)**2 + y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = C_{1} e^{- \int \limits ^{\frac {x}{y{\left (x \right )}}} \frac {u_{1}}{u_{1}^{2} - u_{1} \sqrt {u_{1}^{2} + 2 u_{1} - 3} + u_{1} - 2}\, du_{1} + \int \limits ^{\frac {x}{y{\left (x \right )}}} \frac {\sqrt {u_{1}^{2} + 2 u_{1} - 3}}{u_{1}^{2} - u_{1} \sqrt {u_{1}^{2} + 2 u_{1} - 3} + u_{1} - 2}\, du_{1} - \int \limits ^{\frac {x}{y{\left (x \right )}}} \frac {1}{u_{1}^{2} - u_{1} \sqrt {u_{1}^{2} + 2 u_{1} - 3} + u_{1} - 2}\, du_{1}}, \ y{\left (x \right )} = C_{1} e^{- \int \limits ^{\frac {x}{y{\left (x \right )}}} \frac {u_{1}}{u_{1}^{2} + u_{1} \sqrt {u_{1}^{2} + 2 u_{1} - 3} + u_{1} - 2}\, du_{1} - \int \limits ^{\frac {x}{y{\left (x \right )}}} \frac {\sqrt {u_{1}^{2} + 2 u_{1} - 3}}{u_{1}^{2} + u_{1} \sqrt {u_{1}^{2} + 2 u_{1} - 3} + u_{1} - 2}\, du_{1} - \int \limits ^{\frac {x}{y{\left (x \right )}}} \frac {1}{u_{1}^{2} + u_{1} \sqrt {u_{1}^{2} + 2 u_{1} - 3} + u_{1} - 2}\, du_{1}}\right ]
\]