60.1.556 problem 559
Internal
problem
ID
[10570]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
559
Date
solved
:
Monday, January 27, 2025 at 09:07:54 PM
CAS
classification
:
[[_homogeneous, `class A`], _rational, _dAlembert]
\begin{align*} y \sqrt {1+{y^{\prime }}^{2}}-a y y^{\prime }-a x&=0 \end{align*}
✓ Solution by Maple
Time used: 0.556 (sec). Leaf size: 378
dsolve(y(x)*(diff(y(x),x)^2+1)^(1/2)-a*y(x)*diff(y(x),x)-a*x=0,y(x), singsol=all)
\begin{align*}
-{\mathrm e}^{a \left (\int _{}^{\frac {-a^{2} x +\sqrt {y^{2} \left (a^{2}-1\right )+a^{2} x^{2}}}{\left (a^{2}-1\right ) y}}\frac {a \sqrt {\textit {\_a}^{2}+1}-\textit {\_a}}{\sqrt {\textit {\_a}^{2}+1}\, \left (\textit {\_a} a -\sqrt {\textit {\_a}^{2}+1}\right ) \left (-\sqrt {\textit {\_a}^{2}+1}\, \textit {\_a} +a \left (\textit {\_a}^{2}+1\right )\right )}d \textit {\_a} \right )} c_{1} +x &= 0 \\
-{\mathrm e}^{a \left (\int _{}^{\frac {-a^{2} x -\sqrt {y^{2} \left (a^{2}-1\right )+a^{2} x^{2}}}{y \left (a^{2}-1\right )}}\frac {a \sqrt {\textit {\_a}^{2}+1}-\textit {\_a}}{\sqrt {\textit {\_a}^{2}+1}\, \left (\textit {\_a} a -\sqrt {\textit {\_a}^{2}+1}\right ) \left (-\sqrt {\textit {\_a}^{2}+1}\, \textit {\_a} +a \left (\textit {\_a}^{2}+1\right )\right )}d \textit {\_a} \right )} c_{1} +x &= 0 \\
y &= \operatorname {RootOf}\left (-\ln \left (x \right )+\int _{}^{\textit {\_Z}}-\frac {\left (\textit {\_a}^{2} a^{2}-\textit {\_a}^{2}+a^{2}-\sqrt {\textit {\_a}^{2} a^{2}-\textit {\_a}^{2}+a^{2}}\right ) \textit {\_a}}{\left (\textit {\_a}^{2}+1\right ) \left (\textit {\_a}^{2} a^{2}-\textit {\_a}^{2}+a^{2}\right )}d \textit {\_a} +c_{1} \right ) x \\
y &= \operatorname {RootOf}\left (-\ln \left (x \right )-\int _{}^{\textit {\_Z}}\frac {\left (\textit {\_a}^{2} a^{2}-\textit {\_a}^{2}+a^{2}+\sqrt {\textit {\_a}^{2} a^{2}-\textit {\_a}^{2}+a^{2}}\right ) \textit {\_a}}{\left (\textit {\_a}^{2}+1\right ) \left (\textit {\_a}^{2} a^{2}-\textit {\_a}^{2}+a^{2}\right )}d \textit {\_a} +c_{1} \right ) x \\
\end{align*}
✓ Solution by Mathematica
Time used: 6.795 (sec). Leaf size: 251
DSolve[-(a*x) - a*y[x]*D[y[x],x] + y[x]*Sqrt[1 + D[y[x],x]^2]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to -\frac {\sqrt {\left (a^2-1\right )^3 \left (-x^2\right )-2 \left (a^2-1\right ) x e^{\left (a^2-1\right ) c_1}+e^{2 \left (a^2-1\right ) c_1}}}{\sqrt {\left (a^2-1\right )^3}} \\
y(x)\to \frac {\sqrt {\left (a^2-1\right )^3 \left (-x^2\right )-2 \left (a^2-1\right ) x e^{\left (a^2-1\right ) c_1}+e^{2 \left (a^2-1\right ) c_1}}}{\sqrt {\left (a^2-1\right )^3}} \\
y(x)\to -\frac {\sqrt {\left (a^2-1\right )^3 \left (-x^2\right )+2 \left (a^2-1\right ) x e^{\left (a^2-1\right ) c_1}+e^{2 \left (a^2-1\right ) c_1}}}{\sqrt {\left (a^2-1\right )^3}} \\
y(x)\to \frac {\sqrt {\left (a^2-1\right )^3 \left (-x^2\right )+2 \left (a^2-1\right ) x e^{\left (a^2-1\right ) c_1}+e^{2 \left (a^2-1\right ) c_1}}}{\sqrt {\left (a^2-1\right )^3}} \\
\end{align*}