60.1.543 problem 556
Internal
problem
ID
[10557]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
556
Date
solved
:
Sunday, March 30, 2025 at 06:01:22 PM
CAS
classification
:
[_dAlembert]
\begin{align*} \sqrt {{y^{\prime }}^{2}+1}+x {y^{\prime }}^{2}+y&=0 \end{align*}
✓ Maple. Time used: 0.428 (sec). Leaf size: 693
ode:=(1+diff(y(x),x)^2)^(1/2)+x*diff(y(x),x)^2+y(x) = 0;
dsolve(ode,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+2 \sqrt {4 x^{2}-4 x y+1}+2-4 \sqrt {-4 x y+2 \sqrt {4 x^{2}-4 x y+1}+2}\, x -4 \,\operatorname {arcsinh}\left (\frac {\sqrt {-4 x y+2 \sqrt {4 x^{2}-4 x y+1}+2}}{2 x}\right ) x +c_1 x +4 x^{2}\right )}{{\left (\sqrt {-4 x y+2 \sqrt {4 x^{2}-4 x y+1}+2}-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 \sqrt {4 x^{2}-4 x y+1}+2}+4 x^{2} \operatorname {arcsinh}\left (\frac {\sqrt {-4 x y+2 \sqrt {4 x^{2}-4 x y+1}+2}}{2 x}\right )+2 x \sqrt {4 x^{2}-4 x y+1}+2 x}{{\left (\sqrt {-4 x y+2 \sqrt {4 x^{2}-4 x y+1}+2}+2 x \right )}^{2}} &= 0 \\
-\frac {x \left (4 x \,\operatorname {arcsinh}\left (\frac {\sqrt {-4 x y-2 \sqrt {4 x^{2}-4 x y+1}+2}}{2 x}\right )-2 x \sqrt {\frac {4 x^{2}-4 x y-2 \sqrt {4 x^{2}-4 x y+1}+2}{x^{2}}}+4 x y+2 \sqrt {4 x^{2}-4 x y+1}-2+4 \sqrt {-4 x y-2 \sqrt {4 x^{2}-4 x y+1}+2}\, x -c_1 x -4 x^{2}\right )}{{\left (\sqrt {-4 x y-2 \sqrt {4 x^{2}-4 x y+1}+2}-2 x \right )}^{2}} &= 0 \\
\frac {4 x^{3}+c_1 \,x^{2}-4 x^{2} y+4 x^{2} \sqrt {-4 x y-2 \sqrt {4 x^{2}-4 x y+1}+2}+2 x^{2} \sqrt {\frac {4 x^{2}-4 x y-2 \sqrt {4 x^{2}-4 x y+1}+2}{x^{2}}}+4 x^{2} \operatorname {arcsinh}\left (\frac {\sqrt {-4 x y-2 \sqrt {4 x^{2}-4 x y+1}+2}}{2 x}\right )-2 x \sqrt {4 x^{2}-4 x y+1}+2 x}{{\left (\sqrt {-4 x y-2 \sqrt {4 x^{2}-4 x y+1}+2}+2 x \right )}^{2}} &= 0 \\
\end{align*}
✓ Mathematica. Time used: 3.049 (sec). Leaf size: 78
ode=y[x] + x*D[y[x],x]^2 + Sqrt[1 + D[y[x],x]^2]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [\left \{x=\frac {\log \left (\sqrt {K[1]^2+1}-K[1]\right )-\sqrt {K[1]^2+1}}{(K[1]+1)^2}+\frac {c_1}{(K[1]+1)^2},y(x)=-x K[1]^2-\sqrt {K[1]^2+1}\right \},\{y(x),K[1]\}\right ]
\]
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x*Derivative(y(x), x)**2 + sqrt(Derivative(y(x), x)**2 + 1) + y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out