60.7.43 problem 1640 (6.50)

Internal problem ID [11593]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1640 (6.50)
Date solved : Sunday, March 30, 2025 at 08:30:28 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

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

✓ Maple. Time used: 0.025 (sec). Leaf size: 70

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

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

✓ Mathematica. Time used: 1.11 (sec). Leaf size: 290

ode=b*y[x] + a*y[x]*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 {$\#$1}}-\frac {\sqrt {a}}{\sqrt {e^{2 a c_1-a K[1]^2}-b}}dK[1]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sqrt {a}}{\sqrt {e^{2 a c_1-a K[2]^2}-b}}dK[2]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {\sqrt {a}}{\sqrt {e^{2 a (-c_1)-a K[1]^2}-b}}dK[1]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {\sqrt {a}}{\sqrt {e^{2 a c_1-a K[1]^2}-b}}dK[1]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sqrt {a}}{\sqrt {e^{2 a (-c_1)-a K[2]^2}-b}}dK[2]\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sqrt {a}}{\sqrt {e^{2 a c_1-a K[2]^2}-b}}dK[2]\&\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*y(x)*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 + Derivative(y(x), (x, 2))/y(x))/a) + Derivative(y(x), x) cannot be solved by the factorable group method