60.1.235 problem 236
Internal
problem
ID
[10249]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
236
Date
solved
:
Monday, January 27, 2025 at 06:41:37 PM
CAS
classification
:
[_rational, [_Abel, `2nd type`, `class B`]]
\begin{align*} x \left (y+4\right ) y^{\prime }-y^{2}-2 y-2 x&=0 \end{align*}
✓ Solution by Maple
Time used: 0.001 (sec). Leaf size: 121
dsolve(x*(y(x)+4)*diff(y(x),x)-y(x)^2-2*y(x)-2*x=0,y(x), singsol=all)
\begin{align*}
y &= \frac {-x \sqrt {x +4}\, \sqrt {\frac {\left (x +4\right ) c_{1} -4}{x +4}}-4 \sqrt {x}}{-\sqrt {\frac {\left (x +4\right ) c_{1} -4}{x +4}}\, \sqrt {x +4}+\sqrt {x}} \\
y &= \frac {x \sqrt {x +4}\, \sqrt {\frac {\left (x +4\right ) c_{1} -4}{x +4}}-4 \sqrt {x}}{\sqrt {\frac {\left (x +4\right ) c_{1} -4}{x +4}}\, \sqrt {x +4}+\sqrt {x}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 1.744 (sec). Leaf size: 569
DSolve[x*(y[x]+4)*D[y[x],x]-y[x]^2-2*y[x]-2*x==0,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*}