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
:
Monday, January 27, 2025 at 11:24:26 AM
CAS
classification
:
[[_homogeneous, `class G`], _rational, _dAlembert]
\begin{align*} x {y^{\prime }}^{2}-a y y^{\prime }+b&=0 \end{align*}
✓ Solution by Maple
Time used: 0.063 (sec). Leaf size: 381
dsolve(x*diff(y(x),x)^2-a*y(x)*diff(y(x),x)+b = 0,y(x), singsol=all)
\begin{align*}
\frac {4 \left (a 2^{-\frac {1}{a -1}} y \left (x \right ) \left (a -\frac {1}{2}\right )^{2} \sqrt {a^{2} y \left (x \right )^{2}-4 b x}-\frac {b x 2^{\frac {a -2}{a -1}}}{4}+a \left (a \left (a -\frac {1}{2}\right )^{2} y \left (x \right )^{2}-2 x b \left (a -1\right )\right ) 2^{-\frac {1}{a -1}}\right ) c_{1} {\left (\frac {a y \left (x \right )+\sqrt {a^{2} y \left (x \right )^{2}-4 b x}}{x}\right )}^{\frac {1}{a -1}}+4 x a \left (y \left (x \right ) \left (a -\frac {1}{2}\right ) \sqrt {a^{2} y \left (x \right )^{2}-4 b x}+\left (a^{2}-\frac {1}{2} a \right ) y \left (x \right )^{2}-2 b x \right )}{\left (2 a -1\right ) \left (a y \left (x \right )+\sqrt {a^{2} y \left (x \right )^{2}-4 b x}\right )^{2}} &= 0 \\
\frac {-4 c_{1} \left (-a 2^{-\frac {1}{a -1}} y \left (x \right ) \left (a -\frac {1}{2}\right )^{2} \sqrt {a^{2} y \left (x \right )^{2}-4 b x}-\frac {b x 2^{\frac {a -2}{a -1}}}{4}+a \left (a \left (a -\frac {1}{2}\right )^{2} y \left (x \right )^{2}-2 x b \left (a -1\right )\right ) 2^{-\frac {1}{a -1}}\right ) {\left (\frac {a y \left (x \right )-\sqrt {a^{2} y \left (x \right )^{2}-4 b x}}{x}\right )}^{\frac {1}{a -1}}+4 \left (-y \left (x \right ) \left (a -\frac {1}{2}\right ) \sqrt {a^{2} y \left (x \right )^{2}-4 b x}+\left (a^{2}-\frac {1}{2} a \right ) y \left (x \right )^{2}-2 b x \right ) x a}{\left (2 a -1\right ) \left (a y \left (x \right )-\sqrt {a^{2} y \left (x \right )^{2}-4 b x}\right )^{2}} &= 0 \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.831 (sec). Leaf size: 143
DSolve[x (D[y[x],x])^2-a y[x] D[y[x],x]+b==0,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*}