60.1.456 problem 459
Internal
problem
ID
[10470]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
459
Date
solved
:
Monday, January 27, 2025 at 07:49:32 PM
CAS
classification
:
[[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]
\begin{align*} {\mathrm e}^{-2 x} {y^{\prime }}^{2}-\left (y^{\prime }-1\right )^{2}+{\mathrm e}^{-2 y}&=0 \end{align*}
✓ Solution by Maple
Time used: 0.188 (sec). Leaf size: 128
dsolve(exp(-2*x)*diff(y(x),x)^2-(diff(y(x),x)-1)^2+exp(-2*y(x)) = 0,y(x), singsol=all)
\begin{align*}
y &= c_{1} -\ln \left (\frac {\sqrt {{\mathrm e}^{-2 x +4 c_{1}}-{\mathrm e}^{-2 x +2 c_{1}}}\, {\mathrm e}^{2 x}-{\mathrm e}^{2 c_{1}}}{-{\mathrm e}^{2 c_{1} +2 x}+{\mathrm e}^{2 x}+{\mathrm e}^{2 c_{1}}}\right ) \\
y &= c_{1} -\ln \left (\frac {-\sqrt {{\mathrm e}^{-2 x +4 c_{1}}-{\mathrm e}^{-2 x +2 c_{1}}}\, {\mathrm e}^{2 x}-{\mathrm e}^{2 c_{1}}}{-{\mathrm e}^{2 c_{1} +2 x}+{\mathrm e}^{2 x}+{\mathrm e}^{2 c_{1}}}\right ) \\
\end{align*}
✓ Solution by Mathematica
Time used: 21.066 (sec). Leaf size: 809
DSolve[E^(-2*y[x]) - (-1 + D[y[x],x])^2 + D[y[x],x]^2/E^(2*x)==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
\text {Solve}\left [\int _1^x\left (\frac {\sqrt {e^{2 K[1]}}}{\left (-1+e^{K[1]}\right ) \left (1+e^{K[1]}\right )}-\frac {\sqrt {e^{2 K[1]}} \sqrt {e^{2 K[1]+2 y(x)} \left (-1+e^{2 K[1]}+e^{2 y(x)}\right )}}{\left (-1+e^{K[1]}\right ) \left (1+e^{K[1]}\right ) \left (-1+e^{2 K[1]}+e^{2 y(x)}\right )}\right )dK[1]+\int _1^{y(x)}\left (\frac {\sqrt {e^{2 x+2 K[2]} \left (-1+e^{2 x}+e^{2 K[2]}\right )}}{\sqrt {e^{2 x}} \left (-1+e^{2 x}+e^{2 K[2]}\right )}-\int _1^x\left (\frac {2 e^{2 K[2]} \sqrt {e^{2 K[1]}} \sqrt {e^{2 K[1]+2 K[2]} \left (-1+e^{2 K[1]}+e^{2 K[2]}\right )}}{\left (-1+e^{K[1]}\right ) \left (1+e^{K[1]}\right ) \left (-1+e^{2 K[1]}+e^{2 K[2]}\right )^2}-\frac {\sqrt {e^{2 K[1]}} \left (2 e^{2 K[1]+2 K[2]} \left (-1+e^{2 K[1]}+e^{2 K[2]}\right )+2 e^{2 K[1]+4 K[2]}\right )}{2 \left (-1+e^{K[1]}\right ) \left (1+e^{K[1]}\right ) \left (-1+e^{2 K[1]}+e^{2 K[2]}\right ) \sqrt {e^{2 K[1]+2 K[2]} \left (-1+e^{2 K[1]}+e^{2 K[2]}\right )}}\right )dK[1]\right )dK[2]&=c_1,y(x)\right ] \\
\text {Solve}\left [\int _1^x\left (\frac {\sqrt {e^{2 K[3]+2 y(x)} \left (-1+e^{2 K[3]}+e^{2 y(x)}\right )} \sqrt {e^{2 K[3]}}}{\left (-1+e^{K[3]}\right ) \left (1+e^{K[3]}\right ) \left (-1+e^{2 K[3]}+e^{2 y(x)}\right )}+\frac {\sqrt {e^{2 K[3]}}}{\left (-1+e^{K[3]}\right ) \left (1+e^{K[3]}\right )}\right )dK[3]+\int _1^{y(x)}\left (-\int _1^x\left (\frac {\sqrt {e^{2 K[3]}} \left (2 e^{2 K[3]+2 K[4]} \left (-1+e^{2 K[3]}+e^{2 K[4]}\right )+2 e^{2 K[3]+4 K[4]}\right )}{2 \left (-1+e^{K[3]}\right ) \left (1+e^{K[3]}\right ) \left (-1+e^{2 K[3]}+e^{2 K[4]}\right ) \sqrt {e^{2 K[3]+2 K[4]} \left (-1+e^{2 K[3]}+e^{2 K[4]}\right )}}-\frac {2 e^{2 K[4]} \sqrt {e^{2 K[3]}} \sqrt {e^{2 K[3]+2 K[4]} \left (-1+e^{2 K[3]}+e^{2 K[4]}\right )}}{\left (-1+e^{K[3]}\right ) \left (1+e^{K[3]}\right ) \left (-1+e^{2 K[3]}+e^{2 K[4]}\right )^2}\right )dK[3]-\frac {\sqrt {e^{2 x+2 K[4]} \left (-1+e^{2 x}+e^{2 K[4]}\right )}}{\sqrt {e^{2 x}} \left (-1+e^{2 x}+e^{2 K[4]}\right )}\right )dK[4]&=c_1,y(x)\right ] \\
y(x)\to \log \left (-\sqrt {1-e^{2 x}}\right ) \\
y(x)\to \log \left (\sqrt {1-e^{2 x}}\right ) \\
\end{align*}