60.7.131 problem 1746 (book 6.155)

Internal problem ID [11681]
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 : Sunday, March 30, 2025 at 08:41:53 PM
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: 1.127 (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*} \text {Solution too large to show}\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