54.7.107 problem 1719 (book 6.128)
Internal
problem
ID
[12956]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
6,
non-linear
second
order
Problem
number
:
1719
(book
6.128)
Date
solved
:
Friday, October 03, 2025 at 03:51:02 AM
CAS
classification
:
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]]
\begin{align*} y^{\prime \prime } y+a {y^{\prime }}^{2}+b y y^{\prime }+c y^{2}+d y^{1-a}&=0 \end{align*}
✓ Maple. Time used: 0.084 (sec). Leaf size: 123
ode:=diff(diff(y(x),x),x)*y(x)+a*diff(y(x),x)^2+b*y(x)*diff(y(x),x)+c*y(x)^2+d*y(x)^(-a+1) = 0;
dsolve(ode,y(x), singsol=all);
\[
y = {\mathrm e}^{-\frac {\left (b +\sqrt {\left (-4 a -4\right ) c +b^{2}}\right ) x}{2 a +2}} \left (\frac {\left (-4 a -4\right ) c^{3}+b^{2} c^{2}}{\left (d \,{\mathrm e}^{\frac {\left (b +\sqrt {\left (-4 a -4\right ) c +b^{2}}\right ) x}{2}} \sqrt {\left (-4 a -4\right ) c +b^{2}}+\left (a +1\right ) \left ({\mathrm e}^{x \sqrt {\left (-4 a -4\right ) c +b^{2}}} c_1 -c_2 \right ) c \right )^{2}}\right )^{-\frac {1}{2 a +2}}
\]
✓ Mathematica. Time used: 60.741 (sec). Leaf size: 645
ode=c*y[x]^2 + d*y[x]^(1 - a) + b*y[x]*D[y[x],x] + a*D[y[x],x]^2 + y[x]*D[y[x],{x,2}] == 0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to 2^{\frac {1}{a+1}} \left (\frac {\exp \left (-\frac {x \left (b \sqrt {b^2-4 (a+1) c}-2 (a+1) c+b^2\right )}{\sqrt {b^2-4 (a+1) c}+b}\right ) \left (b^2 \left (-d \sqrt {b^2-4 (a+1) c} \exp \left (\frac {x \left (b \sqrt {b^2-4 (a+1) c}-2 (a+1) c+b^2\right )}{\sqrt {b^2-4 (a+1) c}+b}\right )+c \left (c_2 \sqrt {b^2-4 (a+1) c} \exp \left (\frac {x \left (b \sqrt {b^2-4 (a+1) c}-4 (a+1) c+b^2\right )}{\sqrt {b^2-4 (a+1) c}+b}\right )+a c_1+c_1\right )\right )+(a+1) b c \left (4 d \exp \left (\frac {x \left (b \sqrt {b^2-4 (a+1) c}-2 (a+1) c+b^2\right )}{\sqrt {b^2-4 (a+1) c}+b}\right )-4 c c_2 \exp \left (\frac {x \left (b \sqrt {b^2-4 (a+1) c}-4 (a+1) c+b^2\right )}{\sqrt {b^2-4 (a+1) c}+b}\right )+c_1 \sqrt {b^2-4 (a+1) c}\right )-2 (a+1) c \left (-d \sqrt {b^2-4 (a+1) c} \exp \left (\frac {x \left (b \sqrt {b^2-4 (a+1) c}-2 (a+1) c+b^2\right )}{\sqrt {b^2-4 (a+1) c}+b}\right )+c \left (c_2 \sqrt {b^2-4 (a+1) c} \exp \left (\frac {x \left (b \sqrt {b^2-4 (a+1) c}-4 (a+1) c+b^2\right )}{\sqrt {b^2-4 (a+1) c}+b}\right )+a c_1+c_1\right )\right )+b^3 \left (-d \exp \left (\frac {x \left (b \sqrt {b^2-4 (a+1) c}-2 (a+1) c+b^2\right )}{\sqrt {b^2-4 (a+1) c}+b}\right )+c c_2 \exp \left (\frac {x \left (b \sqrt {b^2-4 (a+1) c}-4 (a+1) c+b^2\right )}{\sqrt {b^2-4 (a+1) c}+b}\right )\right )\right )}{c \left (\sqrt {b^2-4 (a+1) c}+b\right ) \left (b \sqrt {b^2-4 (a+1) c}-4 (a+1) c+b^2\right )}\right ){}^{\frac {1}{a+1}} \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
c = symbols("c")
d = symbols("d")
y = Function("y")
ode = Eq(a*Derivative(y(x), x)**2 + b*y(x)*Derivative(y(x), x) + c*y(x)**2 + d*y(x)**(1 - a) + y(x)*Derivative(y(x), (x, 2)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE Derivative(y(x), x) - (-b*y(x) + sqrt(-4*a*c*y(x)**2 - 4*a*d*y(x)**(1 - a) - 4*a*y(x)*Derivative(y(x), (x, 2)) + b**2*y(x)**2))/(2*a) cannot be solved by the factorable group method