54.7.39 problem 1635 (6.45)

Internal problem ID [12888]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1635 (6.45)
Date solved : Wednesday, October 01, 2025 at 02:44:44 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }+a {y^{\prime }}^{2}+b y&=0 \end{align*}

✓ Maple. Time used: 0.022 (sec). Leaf size: 79

ode:=diff(diff(y(x),x),x)+a*diff(y(x),x)^2+b*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} -2 a \int _{}^{y}\frac {1}{\sqrt {4 \,{\mathrm e}^{-2 a \textit {\_a}} c_1 \,a^{2}-4 b \textit {\_a} a +2 b}}d \textit {\_a} -x -c_2 &= 0 \\ 2 a \int _{}^{y}\frac {1}{\sqrt {4 \,{\mathrm e}^{-2 a \textit {\_a}} c_1 \,a^{2}-4 b \textit {\_a} a +2 b}}d \textit {\_a} -x -c_2 &= 0 \\ \end{align*}

✓ Mathematica. Time used: 0.257 (sec). Leaf size: 332

ode=b*y[x] + a*D[y[x],x]^2 + D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {e^{a K[2]}}{\sqrt {c_1+2 \int _1^{K[2]}-b e^{2 a K[1]} K[1]dK[1]}}dK[2]\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {e^{a K[3]}}{\sqrt {c_1+2 \int _1^{K[3]}-b e^{2 a K[1]} K[1]dK[1]}}dK[3]\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {e^{a K[2]}}{\sqrt {2 \int _1^{K[2]}-b e^{2 a K[1]} K[1]dK[1]-c_1}}dK[2]\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {e^{a K[2]}}{\sqrt {c_1+2 \int _1^{K[2]}-b e^{2 a K[1]} K[1]dK[1]}}dK[2]\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {e^{a K[3]}}{\sqrt {2 \int _1^{K[3]}-b e^{2 a K[1]} K[1]dK[1]-c_1}}dK[3]\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {e^{a K[3]}}{\sqrt {c_1+2 \int _1^{K[3]}-b e^{2 a K[1]} K[1]dK[1]}}dK[3]\&\right ][x+c_2] \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
y = Function("y") 
ode = Eq(a*Derivative(y(x), x)**2 + b*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE sqrt(-(b*y(x) + Derivative(y(x), (x, 2)))/a) + Derivative(y(x), x) cannot be solved by the factorable group method