54.2.118 problem 694
Internal
problem
ID
[11992]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
Additional
non-linear
first
order
Problem
number
:
694
Date
solved
:
Sunday, October 12, 2025 at 02:08:49 AM
CAS
classification
:
[[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]
\begin{align*} y^{\prime }&=\frac {x +1+2 \sqrt {4 x^{2} y+1}\, x^{3}}{2 x^{3} \left (x +1\right )} \end{align*}
✓ Maple. Time used: 0.277 (sec). Leaf size: 36
ode:=diff(y(x),x) = 1/2*(x+1+2*(4*x^2*y(x)+1)^(1/2)*x^3)/x^3/(1+x);
dsolve(ode,y(x), singsol=all);
\[
\frac {-2 \ln \left (x +1\right ) x +c_{1} x +2 x^{2}-\sqrt {4 x^{2} y+1}}{x} = 0
\]
✓ Mathematica. Time used: 0.233 (sec). Leaf size: 640
ode=D[y[x],x] == (1/2 + x/2 + x^3*Sqrt[1 + 4*x^2*y[x]])/(x^3*(1 + x));
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [\int _1^{y(x)}\left (-\frac {\exp \left (-\int _1^x\left (\frac {1}{K[1]+1}-\frac {1}{K[1]}\right )dK[1]\right ) x}{\sqrt {4 K[3] x^2+1}}-\int _1^x\left (-\frac {16 \exp \left (-\int _1^{K[2]}\left (\frac {1}{K[1]+1}-\frac {1}{K[1]}\right )dK[1]\right ) K[3]^2 K[2]^3}{\left (4 K[3] K[2]^2+1\right )^{3/2}}+\frac {4 \exp \left (-\int _1^{K[2]}\left (\frac {1}{K[1]+1}-\frac {1}{K[1]}\right )dK[1]\right ) K[3] K[2]^2}{\left (4 K[3] K[2]^2+1\right )^{3/2}}+\frac {12 \exp \left (-\int _1^{K[2]}\left (\frac {1}{K[1]+1}-\frac {1}{K[1]}\right )dK[1]\right ) K[3] K[2]}{\sqrt {4 K[3] K[2]^2+1}}-\frac {\exp \left (-\int _1^{K[2]}\left (\frac {1}{K[1]+1}-\frac {1}{K[1]}\right )dK[1]\right )}{\sqrt {4 K[3] K[2]^2+1}}-\frac {2 \exp \left (-\int _1^{K[2]}\left (\frac {1}{K[1]+1}-\frac {1}{K[1]}\right )dK[1]\right ) \sqrt {4 K[3] K[2]^2+1}}{K[2]}+\frac {\exp \left (-\int _1^{K[2]}\left (\frac {1}{K[1]+1}-\frac {1}{K[1]}\right )dK[1]\right )}{\sqrt {4 K[3] K[2]^2+1} K[2]}\right )dK[2]-\frac {\exp \left (-\int _1^x\left (\frac {1}{K[1]+1}-\frac {1}{K[1]}\right )dK[1]\right )}{\sqrt {4 K[3] x^2+1}}\right )dK[3]+\int _1^x\left (\frac {8 \exp \left (-\int _1^{K[2]}\left (\frac {1}{K[1]+1}-\frac {1}{K[1]}\right )dK[1]\right ) K[2] y(x)^2}{\sqrt {4 y(x) K[2]^2+1}}-\frac {2 \exp \left (-\int _1^{K[2]}\left (\frac {1}{K[1]+1}-\frac {1}{K[1]}\right )dK[1]\right ) \sqrt {4 y(x) K[2]^2+1} y(x)}{K[2]}-\frac {2 \exp \left (-\int _1^{K[2]}\left (\frac {1}{K[1]+1}-\frac {1}{K[1]}\right )dK[1]\right ) y(x)}{\sqrt {4 y(x) K[2]^2+1}}+\exp \left (-\int _1^{K[2]}\left (\frac {1}{K[1]+1}-\frac {1}{K[1]}\right )dK[1]\right )+\frac {\exp \left (-\int _1^{K[2]}\left (\frac {1}{K[1]+1}-\frac {1}{K[1]}\right )dK[1]\right ) \sqrt {4 y(x) K[2]^2+1}}{2 K[2]^2}+\frac {\exp \left (-\int _1^{K[2]}\left (\frac {1}{K[1]+1}-\frac {1}{K[1]}\right )dK[1]\right ) \sqrt {4 y(x) K[2]^2+1}}{2 K[2]^3}\right )dK[2]=c_1,y(x)\right ]
\]
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(Derivative(y(x), x) - (2*x**3*sqrt(4*x**2*y(x) + 1) + x + 1)/(2*x**3*(x + 1)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out