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
:
Monday, January 27, 2025 at 11:35:31 AM
CAS
classification
:
[[_homogeneous, `class A`], _rational, _dAlembert]
\begin{align*} y {y^{\prime }}^{2}-\left (x +y\right ) y^{\prime }+y&=0 \end{align*}
✓ Solution by Maple
Time used: 0.076 (sec). Leaf size: 176
dsolve(y(x)*diff(y(x),x)^2-(x+y(x))*diff(y(x),x)+y(x) = 0,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*}
✓ Solution by Mathematica
Time used: 2.348 (sec). Leaf size: 192
DSolve[y[x] (D[y[x],x])^2-(x+y[x])D[y[x],x]+y[x]==0,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*}