60.7.188 problem 1810 (book 6.219)
Internal
problem
ID
[11738]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
6,
non-linear
second
order
Problem
number
:
1810
(book
6.219)
Date
solved
:
Thursday, March 13, 2025 at 09:23:09 PM
CAS
classification
:
[NONE]
\begin{align*} \left (c +2 b x +a \,x^{2}+y^{2}\right )^{2} y^{\prime \prime }+d y&=0 \end{align*}
✓ Maple. Time used: 0.138 (sec). Leaf size: 336
ode:=(c+2*b*x+a*x^2+y(x)^2)^2*diff(diff(y(x),x),x)+d*y(x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \operatorname {RootOf}\left (-a \arctan \left (\frac {a x +b}{\sqrt {a c -b^{2}}}\right )+a \left (\int _{}^{\textit {\_Z}}\frac {\sqrt {\left (\textit {\_f}^{2}+1\right ) \left (-\textit {\_f}^{4} a c +\textit {\_f}^{4} b^{2}+c_{1} \textit {\_f}^{2} a^{2}-\textit {\_f}^{2} a c +\textit {\_f}^{2} b^{2}+c_{1} a^{2}+d \right )}}{-\textit {\_f}^{4} a c +\textit {\_f}^{4} b^{2}+c_{1} \textit {\_f}^{2} a^{2}-\textit {\_f}^{2} a c +\textit {\_f}^{2} b^{2}+c_{1} a^{2}+d}d \textit {\_f} \right ) \sqrt {a c -b^{2}}+c_{2} \sqrt {a c -b^{2}}\right ) \sqrt {a \,x^{2}+2 b x +c} \\
y &= \operatorname {RootOf}\left (-a \arctan \left (\frac {a x +b}{\sqrt {a c -b^{2}}}\right )-a \left (\int _{}^{\textit {\_Z}}\frac {\sqrt {\left (\textit {\_f}^{2}+1\right ) \left (-\textit {\_f}^{4} a c +\textit {\_f}^{4} b^{2}+c_{1} \textit {\_f}^{2} a^{2}-\textit {\_f}^{2} a c +\textit {\_f}^{2} b^{2}+c_{1} a^{2}+d \right )}}{-\textit {\_f}^{4} a c +\textit {\_f}^{4} b^{2}+c_{1} \textit {\_f}^{2} a^{2}-\textit {\_f}^{2} a c +\textit {\_f}^{2} b^{2}+c_{1} a^{2}+d}d \textit {\_f} \right ) \sqrt {a c -b^{2}}+c_{2} \sqrt {a c -b^{2}}\right ) \sqrt {a \,x^{2}+2 b x +c} \\
\end{align*}
✓ Mathematica. Time used: 38.051 (sec). Leaf size: 260
ode=d*y[x] + (c + 2*b*x + a*x^2 + y[x]^2)^2*D[y[x],{x,2}] == 0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
\text {Solve}\left [a \arctan \left (\frac {a x+b}{\sqrt {a c-b^2}}\right )+\sqrt {a c-b^2} \int _1^{\frac {y(x)}{\sqrt {c+x (2 b+a x)}}}\frac {a \left (K[2]^2+1\right )}{\sqrt {\left (K[2]^2+1\right ) \left (d+\left (K[2]^2+1\right ) \left (c_1 a^2+\left (b^2-a c\right ) K[2]^2\right )\right )}}dK[2]&=c_2 \sqrt {a c-b^2},y(x)\right ] \\
\text {Solve}\left [a \arctan \left (\frac {a x+b}{\sqrt {a c-b^2}}\right )-\sqrt {a c-b^2} \int _1^{\frac {y(x)}{\sqrt {c+x (2 b+a x)}}}\frac {a \left (K[3]^2+1\right )}{\sqrt {\left (K[3]^2+1\right ) \left (d+\left (K[3]^2+1\right ) \left (c_1 a^2+\left (b^2-a c\right ) K[3]^2\right )\right )}}dK[3]&=c_2 \sqrt {a c-b^2},y(x)\right ] \\
\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(d*y(x) + (a*x**2 + 2*b*x + c + y(x)**2)**2*Derivative(y(x), (x, 2)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : solve: Cannot solve d*y(x) + (a*x**2 + 2*b*x + c + y(x)**2)**2*Derivative(y(x), (x, 2))