31.6.17 problem 17
Internal
problem
ID
[5766]
Book
:
Differential
Equations,
By
George
Boole
F.R.S.
1865
Section
:
Chapter
7
Problem
number
:
17
Date
solved
:
Tuesday, January 28, 2025 at 03:01:01 PM
CAS
classification
:
[[_1st_order, _with_linear_symmetries], _dAlembert]
\begin{align*} x +y^{\prime } y&=a \sqrt {1+{y^{\prime }}^{2}} \end{align*}
✓ Solution by Maple
Time used: 0.161 (sec). Leaf size: 237
dsolve(x+y(x)*diff(y(x),x)=a*sqrt(1+(diff(y(x),x))^2),y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= \csc \left (\operatorname {RootOf}\left (\left (\sin \left (\textit {\_Z} \right ) \textit {\_Z} a -\cos \left (\textit {\_Z} \right ) a +\sin \left (\textit {\_Z} \right ) c_{1} -x \right ) \left (\sin \left (\textit {\_Z} \right ) \textit {\_Z} a +\cos \left (\textit {\_Z} \right ) a +\sin \left (\textit {\_Z} \right ) c_{1} -x \right )\right )\right ) \operatorname {csgn}\left (\sec \left (\operatorname {RootOf}\left (\left (\sin \left (\textit {\_Z} \right ) \textit {\_Z} a -\cos \left (\textit {\_Z} \right ) a +\sin \left (\textit {\_Z} \right ) c_{1} -x \right ) \left (\sin \left (\textit {\_Z} \right ) \textit {\_Z} a +\cos \left (\textit {\_Z} \right ) a +\sin \left (\textit {\_Z} \right ) c_{1} -x \right )\right )\right )\right ) a -\cot \left (\operatorname {RootOf}\left (\left (\sin \left (\textit {\_Z} \right ) \textit {\_Z} a -\cos \left (\textit {\_Z} \right ) a +\sin \left (\textit {\_Z} \right ) c_{1} -x \right ) \left (\sin \left (\textit {\_Z} \right ) \textit {\_Z} a +\cos \left (\textit {\_Z} \right ) a +\sin \left (\textit {\_Z} \right ) c_{1} -x \right )\right )\right ) x \\
y \left (x \right ) &= \csc \left (\operatorname {RootOf}\left (\left (\sin \left (\textit {\_Z} \right ) \textit {\_Z} a +\cos \left (\textit {\_Z} \right ) a +\sin \left (\textit {\_Z} \right ) c_{1} +x \right ) \left (\sin \left (\textit {\_Z} \right ) \textit {\_Z} a -\cos \left (\textit {\_Z} \right ) a +\sin \left (\textit {\_Z} \right ) c_{1} +x \right )\right )\right ) \operatorname {csgn}\left (\sec \left (\operatorname {RootOf}\left (\left (\sin \left (\textit {\_Z} \right ) \textit {\_Z} a +\cos \left (\textit {\_Z} \right ) a +\sin \left (\textit {\_Z} \right ) c_{1} +x \right ) \left (\sin \left (\textit {\_Z} \right ) \textit {\_Z} a -\cos \left (\textit {\_Z} \right ) a +\sin \left (\textit {\_Z} \right ) c_{1} +x \right )\right )\right )\right ) a -\cot \left (\operatorname {RootOf}\left (\left (\sin \left (\textit {\_Z} \right ) \textit {\_Z} a +\cos \left (\textit {\_Z} \right ) a +\sin \left (\textit {\_Z} \right ) c_{1} +x \right ) \left (\sin \left (\textit {\_Z} \right ) \textit {\_Z} a -\cos \left (\textit {\_Z} \right ) a +\sin \left (\textit {\_Z} \right ) c_{1} +x \right )\right )\right ) x \\
\end{align*}
✓ Solution by Mathematica
Time used: 3.822 (sec). Leaf size: 150
DSolve[x+y[x]*D[y[x],x]==a*Sqrt[1+(D[y[x],x])^2],y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
\text {Solve}\left [\frac {a^2 \arctan \left (\frac {\sqrt {a^2 \left (-a^2+x^2+y(x)^2\right )}}{a^2}\right )+a^2 \arctan \left (\frac {x}{y(x)}\right )-\sqrt {a^2 \left (-a^2+x^2+y(x)^2\right )}}{a^2}&=c_1,y(x)\right ] \\
\text {Solve}\left [\frac {-a^2 \arctan \left (\frac {\sqrt {a^2 \left (-a^2+x^2+y(x)^2\right )}}{a^2}\right )+a^2 \arctan \left (\frac {x}{y(x)}\right )+\sqrt {a^2 \left (-a^2+x^2+y(x)^2\right )}}{a^2}&=c_1,y(x)\right ] \\
\end{align*}