54.7.131 problem 1746 (book 6.155)
Internal
problem
ID
[12980]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
6,
non-linear
second
order
Problem
number
:
1746
(book
6.155)
Date
solved
:
Wednesday, October 01, 2025 at 02:55:36 AM
CAS
classification
:
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
\begin{align*} 2 \left (y-a \right ) y^{\prime \prime }+{y^{\prime }}^{2}+1&=0 \end{align*}
✓ Maple. Time used: 0.364 (sec). Leaf size: 123
ode:=2*(y(x)-a)*diff(diff(y(x),x),x)+diff(y(x),x)^2+1 = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
-\sqrt {-\left (-y+a \right ) \left (-y+c_1 +a \right )}+\frac {\arctan \left (\frac {2 y-2 a -c_1}{2 \sqrt {-\left (-y+a \right ) \left (-y+c_1 +a \right )}}\right ) c_1}{2}-x -c_2 &= 0 \\
\sqrt {-\left (-y+a \right ) \left (-y+c_1 +a \right )}-\frac {\arctan \left (\frac {2 y-2 a -c_1}{2 \sqrt {-\left (-y+a \right ) \left (-y+c_1 +a \right )}}\right ) c_1}{2}-x -c_2 &= 0 \\
\end{align*}
✓ Mathematica. Time used: 0.755 (sec). Leaf size: 775
ode=1 + D[y[x],x]^2 + 2*(-a + y[x])*D[y[x],{x,2}] == 0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \text {InverseFunction}\left [-\frac {2 \sqrt {a-\text {$\#$1}} \left (2 \text {$\#$1}-2 a+e^{2 c_1}\right )-\sqrt {2} e^{3 c_1} \sqrt {e^{-2 c_1} \left (2 \text {$\#$1}-2 a+e^{2 c_1}\right )} \arcsin \left (\sqrt {2} e^{-c_1} \sqrt {a-\text {$\#$1}}\right )}{2 \sqrt {2} \sqrt {2 \text {$\#$1}-2 a+e^{2 c_1}}}\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [\frac {2 \sqrt {a-\text {$\#$1}} \left (2 \text {$\#$1}-2 a+e^{2 c_1}\right )-\sqrt {2} e^{3 c_1} \sqrt {e^{-2 c_1} \left (2 \text {$\#$1}-2 a+e^{2 c_1}\right )} \arcsin \left (\sqrt {2} e^{-c_1} \sqrt {a-\text {$\#$1}}\right )}{2 \sqrt {2} \sqrt {2 \text {$\#$1}-2 a+e^{2 c_1}}}\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [-\frac {2 \sqrt {a-\text {$\#$1}} \left (2 \text {$\#$1}-2 a+e^{2 (-c_1)}\right )-\sqrt {2} e^{3 (-c_1)} \sqrt {e^{-2 (-c_1)} \left (2 \text {$\#$1}-2 a+e^{2 (-c_1)}\right )} \arcsin \left (\sqrt {2} e^{-(-c_1)} \sqrt {a-\text {$\#$1}}\right )}{2 \sqrt {2} \sqrt {2 \text {$\#$1}-2 a+e^{2 (-c_1)}}}\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [\frac {2 \sqrt {a-\text {$\#$1}} \left (2 \text {$\#$1}-2 a+e^{2 (-c_1)}\right )-\sqrt {2} e^{3 (-c_1)} \sqrt {e^{-2 (-c_1)} \left (2 \text {$\#$1}-2 a+e^{2 (-c_1)}\right )} \arcsin \left (\sqrt {2} e^{-(-c_1)} \sqrt {a-\text {$\#$1}}\right )}{2 \sqrt {2} \sqrt {2 \text {$\#$1}-2 a+e^{2 (-c_1)}}}\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [-\frac {2 \sqrt {a-\text {$\#$1}} \left (2 \text {$\#$1}-2 a+e^{2 c_1}\right )-\sqrt {2} e^{3 c_1} \sqrt {e^{-2 c_1} \left (2 \text {$\#$1}-2 a+e^{2 c_1}\right )} \arcsin \left (\sqrt {2} e^{-c_1} \sqrt {a-\text {$\#$1}}\right )}{2 \sqrt {2} \sqrt {2 \text {$\#$1}-2 a+e^{2 c_1}}}\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [\frac {2 \sqrt {a-\text {$\#$1}} \left (2 \text {$\#$1}-2 a+e^{2 c_1}\right )-\sqrt {2} e^{3 c_1} \sqrt {e^{-2 c_1} \left (2 \text {$\#$1}-2 a+e^{2 c_1}\right )} \arcsin \left (\sqrt {2} e^{-c_1} \sqrt {a-\text {$\#$1}}\right )}{2 \sqrt {2} \sqrt {2 \text {$\#$1}-2 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((-2*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(2*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