77.1.84 problem 111 (page 172)
Internal
problem
ID
[17903]
Book
:
V.V.
Stepanov,
A
course
of
differential
equations
(in
Russian),
GIFML.
Moscow
(1958)
Section
:
All
content
Problem
number
:
111
(page
172)
Date
solved
:
Monday, March 31, 2025 at 04:49:38 PM
CAS
classification
:
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
\begin{align*} 2 \left (2 a -y\right ) y^{\prime \prime }&=1+{y^{\prime }}^{2} \end{align*}
✓ Maple. Time used: 0.363 (sec). Leaf size: 139
ode:=2*(2*a-y(x))*diff(diff(y(x),x),x) = 1+diff(y(x),x)^2;
dsolve(ode,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*}
✓ Mathematica. Time used: 1.215 (sec). Leaf size: 520
ode=2*(2*a-y[x])*D[y[x],{x,2}]== 1+D[y[x],x]^2;
ic={};
DSolve[{ode,ic},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*}
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
y = Function("y")
ode = Eq((4*a - 2*y(x))*Derivative(y(x), (x, 2)) - Derivative(y(x), x)**2 - 1,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE -sqrt(4*a*Derivative(y(x), (x, 2)) - 2*y(x)*Derivative(y(x), (x, 2)) - 1) + Derivative(y(x), x) cannot be solved by the factorable group method