60.1.553 problem 556
Internal
problem
ID
[10567]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
556
Date
solved
:
Monday, January 27, 2025 at 09:05:48 PM
CAS
classification
:
[_dAlembert]
\begin{align*} \sqrt {1+{y^{\prime }}^{2}}+x {y^{\prime }}^{2}+y&=0 \end{align*}
✓ Solution by Maple
Time used: 0.167 (sec). Leaf size: 683
dsolve((diff(y(x),x)^2+1)^(1/2)+x*diff(y(x),x)^2+y(x)=0,y(x), singsol=all)
\begin{align*}
y &= -1 \\
-\frac {x \left (-2 \sqrt {2}\, \sqrt {\frac {2 x^{2}-2 x y+\sqrt {4 x^{2}-4 x y+1}+1}{x^{2}}}\, x +4 x y+4 x \sqrt {-4 x y+2+2 \sqrt {4 x^{2}-4 x y+1}}+4 \,\operatorname {arcsinh}\left (\frac {\sqrt {-4 x y+2+2 \sqrt {4 x^{2}-4 x y+1}}}{2 x}\right ) x -c_{1} x -4 x^{2}-2 \sqrt {4 x^{2}-4 x y+1}-2\right )}{{\left (\sqrt {-4 x y+2+2 \sqrt {4 x^{2}-4 x y+1}}-2 x \right )}^{2}} &= 0 \\
\frac {2 x^{2} \sqrt {2}\, \sqrt {\frac {2 x^{2}-2 x y+\sqrt {4 x^{2}-4 x y+1}+1}{x^{2}}}+4 x^{3}+c_{1} x^{2}-4 x^{2} y+4 x^{2} \sqrt {-4 x y+2+2 \sqrt {4 x^{2}-4 x y+1}}+4 x^{2} \operatorname {arcsinh}\left (\frac {\sqrt {-4 x y+2+2 \sqrt {4 x^{2}-4 x y+1}}}{2 x}\right )+2 x \sqrt {4 x^{2}-4 x y+1}+2 x}{{\left (\sqrt {-4 x y+2+2 \sqrt {4 x^{2}-4 x y+1}}+2 x \right )}^{2}} &= 0 \\
\frac {2 x \left (-2 x \sqrt {-4 x y-2 \sqrt {4 x^{2}-4 x y+1}+2}-\sqrt {4 x^{2}-4 x y+1}-2 x y+1+2 x^{2}+\left (\sqrt {\frac {4 x^{2}-4 x y-2 \sqrt {4 x^{2}-4 x y+1}+2}{x^{2}}}-2 \,\operatorname {arcsinh}\left (\frac {\sqrt {-4 x y-2 \sqrt {4 x^{2}-4 x y+1}+2}}{2 x}\right )+\frac {c_{1}}{2}\right ) x \right )}{{\left (-\sqrt {-4 x y-2 \sqrt {4 x^{2}-4 x y+1}+2}+2 x \right )}^{2}} &= 0 \\
\frac {2 \left (2 x \sqrt {-4 x y-2 \sqrt {4 x^{2}-4 x y+1}+2}-\sqrt {4 x^{2}-4 x y+1}-2 x y+1+2 x^{2}+\left (\sqrt {\frac {4 x^{2}-4 x y-2 \sqrt {4 x^{2}-4 x y+1}+2}{x^{2}}}+2 \,\operatorname {arcsinh}\left (\frac {\sqrt {-4 x y-2 \sqrt {4 x^{2}-4 x y+1}+2}}{2 x}\right )+\frac {c_{1}}{2}\right ) x \right ) x}{{\left (\sqrt {-4 x y-2 \sqrt {4 x^{2}-4 x y+1}+2}+2 x \right )}^{2}} &= 0 \\
\end{align*}
✓ Solution by Mathematica
Time used: 3.049 (sec). Leaf size: 78
DSolve[y[x] + x*D[y[x],x]^2 + Sqrt[1 + D[y[x],x]^2]==0,y[x],x,IncludeSingularSolutions -> True]
\[
\text {Solve}\left [\left \{x=\frac {\log \left (\sqrt {K[1]^2+1}-K[1]\right )-\sqrt {K[1]^2+1}}{(K[1]+1)^2}+\frac {c_1}{(K[1]+1)^2},y(x)=-x K[1]^2-\sqrt {K[1]^2+1}\right \},\{y(x),K[1]\}\right ]
\]