77.1.95 problem 122 (page 179)
Internal
problem
ID
[17985]
Book
:
V.V.
Stepanov,
A
course
of
differential
equations
(in
Russian),
GIFML.
Moscow
(1958)
Section
:
All
content
Problem
number
:
122
(page
179)
Date
solved
:
Tuesday, January 28, 2025 at 11:19:27 AM
CAS
classification
:
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
\begin{align*} y y^{\prime \prime }-{y^{\prime }}^{2}-{y^{\prime }}^{4}&=0 \end{align*}
✓ Solution by Maple
Time used: 0.185 (sec). Leaf size: 105
dsolve(y(x)*diff(y(x),x$2)-diff(y(x),x)^2-diff(y(x),x)^4=0,y(x), singsol=all)
\begin{align*}
y &= 0 \\
c_{1} \ln \left (2\right )+c_{1} \ln \left (\frac {c_{1} \left (\sqrt {c_{1}^{2}-y^{2}}+c_{1} \right )}{y}\right )-\sqrt {c_{1}^{2}-y^{2}}-c_{2} -x &= 0 \\
-c_{1} \ln \left (2\right )-c_{1} \ln \left (\frac {c_{1} \left (\sqrt {c_{1}^{2}-y^{2}}+c_{1} \right )}{y}\right )+\sqrt {c_{1}^{2}-y^{2}}-c_{2} -x &= 0 \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.858 (sec). Leaf size: 361
DSolve[y[x]*D[y[x],{x,2}]-D[y[x],x]^2-D[y[x],x]^4==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \text {InverseFunction}\left [-i e^{-c_1} \left (\sqrt {-1+\text {$\#$1}^2 e^{2 c_1}}-\arctan \left (\sqrt {-1+\text {$\#$1}^2 e^{2 c_1}}\right )\right )\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [i e^{-c_1} \left (\sqrt {-1+\text {$\#$1}^2 e^{2 c_1}}-\arctan \left (\sqrt {-1+\text {$\#$1}^2 e^{2 c_1}}\right )\right )\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [-i e^{-(-c_1)} \left (\sqrt {-1+\text {$\#$1}^2 e^{2 (-c_1)}}-\arctan \left (\sqrt {-1+\text {$\#$1}^2 e^{2 (-c_1)}}\right )\right )\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [i e^{-(-c_1)} \left (\sqrt {-1+\text {$\#$1}^2 e^{2 (-c_1)}}-\arctan \left (\sqrt {-1+\text {$\#$1}^2 e^{2 (-c_1)}}\right )\right )\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [-i e^{-c_1} \left (\sqrt {-1+\text {$\#$1}^2 e^{2 c_1}}-\arctan \left (\sqrt {-1+\text {$\#$1}^2 e^{2 c_1}}\right )\right )\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [i e^{-c_1} \left (\sqrt {-1+\text {$\#$1}^2 e^{2 c_1}}-\arctan \left (\sqrt {-1+\text {$\#$1}^2 e^{2 c_1}}\right )\right )\&\right ][x+c_2] \\
\end{align*}