29.28.19 problem 817
Internal
problem
ID
[5400]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
28
Problem
number
:
817
Date
solved
:
Monday, January 27, 2025 at 11:18:57 AM
CAS
classification
:
[_dAlembert]
\begin{align*} {y^{\prime }}^{2}+a y y^{\prime }-a x&=0 \end{align*}
✓ Solution by Maple
Time used: 0.051 (sec). Leaf size: 398
dsolve(diff(y(x),x)^2+a*y(x)*diff(y(x),x)-a*x = 0,y(x), singsol=all)
\begin{align*}
\frac {\left (-a y \left (x \right )+\sqrt {a \left (y \left (x \right )^{2} a +4 x \right )}\right ) c_{1}}{\sqrt {-2 a y \left (x \right )+2 \sqrt {a \left (y \left (x \right )^{2} a +4 x \right )}-4}\, \sqrt {-2 a y \left (x \right )+2 \sqrt {a \left (y \left (x \right )^{2} a +4 x \right )}+4}}+x +\frac {\left (-a y \left (x \right )+\sqrt {a \left (y \left (x \right )^{2} a +4 x \right )}\right ) \left (-\ln \left (2\right )+\ln \left (-a y \left (x \right )+\sqrt {a \left (y \left (x \right )^{2} a +4 x \right )}+\sqrt {2 a^{2} y \left (x \right )^{2}-2 \sqrt {a \left (y \left (x \right )^{2} a +4 x \right )}\, a y \left (x \right )+4 a x -4}\right )\right )}{a \sqrt {2 a^{2} y \left (x \right )^{2}-2 \sqrt {a \left (y \left (x \right )^{2} a +4 x \right )}\, a y \left (x \right )+4 a x -4}} &= 0 \\
\frac {\left (a y \left (x \right )+\sqrt {a \left (y \left (x \right )^{2} a +4 x \right )}\right ) c_{1}}{\sqrt {-2 a y \left (x \right )-2 \sqrt {a \left (y \left (x \right )^{2} a +4 x \right )}-4}\, \sqrt {-2 a y \left (x \right )-2 \sqrt {a \left (y \left (x \right )^{2} a +4 x \right )}+4}}+x -\frac {\left (a y \left (x \right )+\sqrt {a \left (y \left (x \right )^{2} a +4 x \right )}\right ) \left (-\ln \left (2\right )+\ln \left (-a y \left (x \right )-\sqrt {a \left (y \left (x \right )^{2} a +4 x \right )}+\sqrt {2 a^{2} y \left (x \right )^{2}+2 \sqrt {a \left (y \left (x \right )^{2} a +4 x \right )}\, a y \left (x \right )+4 a x -4}\right )\right )}{a \sqrt {2 a^{2} y \left (x \right )^{2}+2 \sqrt {a \left (y \left (x \right )^{2} a +4 x \right )}\, a y \left (x \right )+4 a x -4}} &= 0 \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.914 (sec). Leaf size: 65
DSolve[(D[y[x],x])^2+a*y[x]*D[y[x],x]-a*x==0,y[x],x,IncludeSingularSolutions -> True]
\[
\text {Solve}\left [\left \{x=\frac {K[1] \arcsin (K[1])}{a \sqrt {1-K[1]^2}}+\frac {c_1 K[1]}{\sqrt {1-K[1]^2}},y(x)=\frac {x}{K[1]}-\frac {K[1]}{a}\right \},\{y(x),K[1]\}\right ]
\]