77.1.84 problem 111 (page 172)
Internal
problem
ID
[17974]
Book
:
V.V.
Stepanov,
A
course
of
differential
equations
(in
Russian),
GIFML.
Moscow
(1958)
Section
:
All
content
Problem
number
:
111
(page
172)
Date
solved
:
Tuesday, January 28, 2025 at 11:19:11 AM
CAS
classification
:
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
\begin{align*} 2 \left (2 a -y\right ) y^{\prime \prime }&={y^{\prime }}^{2}+1 \end{align*}
✓ Solution by Maple
Time used: 0.195 (sec). Leaf size: 139
dsolve(2*(2*a-y(x))*diff(y(x),x$2)= 1+diff(y(x),x)^2,y(x), singsol=all)
\begin{align*}
-\sqrt {-\left (2 a -y\right ) \left (-y+c_{1} +2 a \right )}+\frac {\arctan \left (\frac {2 y-4 a -c_{1}}{2 \sqrt {-\left (2 a -y\right ) \left (-y+c_{1} +2 a \right )}}\right ) c_{1}}{2}-x -c_{2} &= 0 \\
\sqrt {-\left (2 a -y\right ) \left (-y+c_{1} +2 a \right )}-\frac {\arctan \left (\frac {2 y-4 a -c_{1}}{2 \sqrt {-\left (2 a -y\right ) \left (-y+c_{1} +2 a \right )}}\right ) c_{1}}{2}-x -c_{2} &= 0 \\
\end{align*}
✓ Solution by Mathematica
Time used: 1.215 (sec). Leaf size: 520
DSolve[2*(2*a-y[x])*D[y[x],{x,2}]== 1+D[y[x],x]^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \text {InverseFunction}\left [-\frac {1}{2} e^{2 c_1} \arctan \left (\frac {\sqrt {2 \text {$\#$1}-4 a+e^{2 c_1}}}{\sqrt {4 a-2 \text {$\#$1}}}\right )-\sqrt {a-\frac {\text {$\#$1}}{2}} \sqrt {2 \text {$\#$1}-4 a+e^{2 c_1}}\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\frac {1}{2} e^{2 c_1} \arctan \left (\frac {\sqrt {2 \text {$\#$1}-4 a+e^{2 c_1}}}{\sqrt {4 a-2 \text {$\#$1}}}\right )+\sqrt {a-\frac {\text {$\#$1}}{2}} \sqrt {2 \text {$\#$1}-4 a+e^{2 c_1}}\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [-\frac {1}{2} e^{2 (-c_1)} \arctan \left (\frac {\sqrt {2 \text {$\#$1}-4 a+e^{2 (-c_1)}}}{\sqrt {4 a-2 \text {$\#$1}}}\right )-\sqrt {a-\frac {\text {$\#$1}}{2}} \sqrt {2 \text {$\#$1}-4 a+e^{2 (-c_1)}}\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\frac {1}{2} e^{2 (-c_1)} \arctan \left (\frac {\sqrt {2 \text {$\#$1}-4 a+e^{2 (-c_1)}}}{\sqrt {4 a-2 \text {$\#$1}}}\right )+\sqrt {a-\frac {\text {$\#$1}}{2}} \sqrt {2 \text {$\#$1}-4 a+e^{2 (-c_1)}}\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [-\frac {1}{2} e^{2 c_1} \arctan \left (\frac {\sqrt {2 \text {$\#$1}-4 a+e^{2 c_1}}}{\sqrt {4 a-2 \text {$\#$1}}}\right )-\sqrt {a-\frac {\text {$\#$1}}{2}} \sqrt {2 \text {$\#$1}-4 a+e^{2 c_1}}\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\frac {1}{2} e^{2 c_1} \arctan \left (\frac {\sqrt {2 \text {$\#$1}-4 a+e^{2 c_1}}}{\sqrt {4 a-2 \text {$\#$1}}}\right )+\sqrt {a-\frac {\text {$\#$1}}{2}} \sqrt {2 \text {$\#$1}-4 a+e^{2 c_1}}\&\right ][x+c_2] \\
\end{align*}