23.1.536 problem 526
Internal
problem
ID
[5143]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Part
II.
Chapter
1.
THE
DIFFERENTIAL
EQUATION
IS
OF
FIRST
ORDER
AND
OF
FIRST
DEGREE,
page
223
Problem
number
:
526
Date
solved
:
Tuesday, September 30, 2025 at 11:44:15 AM
CAS
classification
:
[_rational, [_Abel, `2nd type`, `class B`]]
\begin{align*} x \left (4+y\right ) y^{\prime }&=2 x +2 y+y^{2} \end{align*}
✓ Maple. Time used: 0.003 (sec). Leaf size: 125
ode:=x*(4+y(x))*diff(y(x),x) = 2*x+2*y(x)+y(x)^2;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= x \\
y &= \frac {-\sqrt {4+x}\, \sqrt {\frac {c_1 \left (4+x \right )-4}{4+x}}\, x -4 \sqrt {x}}{-\sqrt {4+x}\, \sqrt {\frac {c_1 \left (4+x \right )-4}{4+x}}+\sqrt {x}} \\
y &= \frac {\sqrt {4+x}\, \sqrt {\frac {c_1 \left (4+x \right )-4}{4+x}}\, x -4 \sqrt {x}}{\sqrt {4+x}\, \sqrt {\frac {c_1 \left (4+x \right )-4}{4+x}}+\sqrt {x}} \\
\end{align*}
✓ Mathematica. Time used: 1.763 (sec). Leaf size: 569
ode=x*(4+y[x])*D[y[x],x]==2*x+2*y[x]+y[x]^2;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to -4+\frac {1}{x \left (\frac {1}{x^2+4 x}-\frac {\exp \left (\int _1^x-\frac {2 (K[1]+1)}{K[1] (K[1]+4)}dK[1]\right )}{\sqrt {-2 \int _1^x-2 \exp \left (2 \int _1^{K[2]}-\frac {2 (K[1]+1)}{K[1] (K[1]+4)}dK[1]\right ) K[2] (K[2]+4)dK[2]+c_1}}\right )}\\ y(x)&\to -4+\frac {1}{x \left (\frac {1}{x^2+4 x}+\frac {\exp \left (\int _1^x-\frac {2 (K[1]+1)}{K[1] (K[1]+4)}dK[1]\right )}{\sqrt {-2 \int _1^x-2 \exp \left (2 \int _1^{K[2]}-\frac {2 (K[1]+1)}{K[1] (K[1]+4)}dK[1]\right ) K[2] (K[2]+4)dK[2]+c_1}}\right )}\\ y(x)&\to x\\ y(x)&\to -\frac {2 x \left (2 \sqrt {2} (x+4) \exp \left (\int _1^x-\frac {2 (K[1]+1)}{K[1] (K[1]+4)}dK[1]\right )+\sqrt {-\int _1^x-2 \exp \left (2 \int _1^{K[2]}-\frac {2 (K[1]+1)}{K[1] (K[1]+4)}dK[1]\right ) K[2] (K[2]+4)dK[2]}\right )}{\sqrt {2} x (x+4) \exp \left (\int _1^x-\frac {2 (K[1]+1)}{K[1] (K[1]+4)}dK[1]\right )-2 \sqrt {-\int _1^x-2 \exp \left (2 \int _1^{K[2]}-\frac {2 (K[1]+1)}{K[1] (K[1]+4)}dK[1]\right ) K[2] (K[2]+4)dK[2]}}\\ y(x)&\to \frac {2 x \left (\sqrt {-\int _1^x-2 \exp \left (2 \int _1^{K[2]}-\frac {2 (K[1]+1)}{K[1] (K[1]+4)}dK[1]\right ) K[2] (K[2]+4)dK[2]}-2 \sqrt {2} (x+4) \exp \left (\int _1^x-\frac {2 (K[1]+1)}{K[1] (K[1]+4)}dK[1]\right )\right )}{\sqrt {2} x (x+4) \exp \left (\int _1^x-\frac {2 (K[1]+1)}{K[1] (K[1]+4)}dK[1]\right )+2 \sqrt {-\int _1^x-2 \exp \left (2 \int _1^{K[2]}-\frac {2 (K[1]+1)}{K[1] (K[1]+4)}dK[1]\right ) K[2] (K[2]+4)dK[2]}} \end{align*}
✓ Sympy. Time used: 3.788 (sec). Leaf size: 68
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x*(y(x) + 4)*Derivative(y(x), x) - 2*x - y(x)**2 - 2*y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = \frac {\left (2 x e^{C_{1}} + x - \sqrt {x \left (2 x e^{C_{1}} + x + 8 e^{C_{1}}\right )}\right ) e^{- C_{1}}}{2}, \ y{\left (x \right )} = \frac {\left (2 x e^{C_{1}} + x + \sqrt {x \left (2 x e^{C_{1}} + x + 8 e^{C_{1}}\right )}\right ) e^{- C_{1}}}{2}\right ]
\]