60.1.555 problem 558
Internal
problem
ID
[10569]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
558
Date
solved
:
Monday, January 27, 2025 at 09:07:08 PM
CAS
classification
:
[[_homogeneous, `class A`], _dAlembert]
\begin{align*} a x \sqrt {1+{y^{\prime }}^{2}}+x y^{\prime }-y&=0 \end{align*}
✓ Solution by Maple
Time used: 0.098 (sec). Leaf size: 340
dsolve(a*x*(diff(y(x),x)^2+1)^(1/2)+x*diff(y(x),x)-y(x)=0,y(x), singsol=all)
\begin{align*}
\frac {x \sqrt {\frac {-a^{2} x^{2}+a^{2} y^{2}+2 \sqrt {-a^{2} x^{2}+x^{2}+y^{2}}\, a y+x^{2}+y^{2}}{\left (a^{2}-1\right )^{2} x^{2}}}-{\mathrm e}^{\frac {\operatorname {arcsinh}\left (\frac {\sqrt {-a^{2} x^{2}+x^{2}+y^{2}}\, a +y}{\left (a^{2}-1\right ) x}\right )}{a}} c_{1}}{\sqrt {\frac {-a^{2} x^{2}+a^{2} y^{2}+2 \sqrt {-a^{2} x^{2}+x^{2}+y^{2}}\, a y+x^{2}+y^{2}}{\left (a^{2}-1\right )^{2} x^{2}}}} &= 0 \\
\frac {x \sqrt {\frac {-a^{2} x^{2}+a^{2} y^{2}-2 \sqrt {-a^{2} x^{2}+x^{2}+y^{2}}\, a y+x^{2}+y^{2}}{\left (a^{2}-1\right )^{2} x^{2}}}-{\mathrm e}^{\frac {\operatorname {arcsinh}\left (\frac {-\sqrt {-a^{2} x^{2}+x^{2}+y^{2}}\, a +y}{\left (a^{2}-1\right ) x}\right )}{a}} c_{1}}{\sqrt {\frac {-a^{2} x^{2}+a^{2} y^{2}-2 \sqrt {-a^{2} x^{2}+x^{2}+y^{2}}\, a y+x^{2}+y^{2}}{\left (a^{2}-1\right )^{2} x^{2}}}} &= 0 \\
\end{align*}
✓ Solution by Mathematica
Time used: 1.687 (sec). Leaf size: 223
DSolve[-y[x] + x*D[y[x],x] + a*x*Sqrt[1 + D[y[x],x]^2]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
\text {Solve}\left [\frac {2 i \arctan \left (\frac {y(x)}{x \sqrt {a^2-\frac {y(x)^2}{x^2}-1}}\right )-2 i a \arctan \left (\frac {a y(x)}{x \sqrt {a^2-\frac {y(x)^2}{x^2}-1}}\right )+a \log \left (\frac {y(x)^2}{x^2}+1\right )}{2 a^2-2}&=\frac {a \log \left (x-a^2 x\right )}{1-a^2}+c_1,y(x)\right ] \\
\text {Solve}\left [\frac {-2 i \arctan \left (\frac {y(x)}{x \sqrt {a^2-\frac {y(x)^2}{x^2}-1}}\right )+2 i a \arctan \left (\frac {a y(x)}{x \sqrt {a^2-\frac {y(x)^2}{x^2}-1}}\right )+a \log \left (\frac {y(x)^2}{x^2}+1\right )}{2 a^2-2}&=\frac {a \log \left (x-a^2 x\right )}{1-a^2}+c_1,y(x)\right ] \\
\end{align*}