60.2.137 problem 713
Internal
problem
ID
[10711]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
Additional
non-linear
first
order
Problem
number
:
713
Date
solved
:
Wednesday, March 05, 2025 at 12:26:42 PM
CAS
classification
:
[[_1st_order, _with_linear_symmetries], _rational, [_Abel, `2nd type`, `class A`]]
\begin{align*} y^{\prime }&=\frac {-b y a +b^{2}+a b +b^{2} x -b a \sqrt {x}-a^{2}}{a \left (-a y+b +a +b x -a \sqrt {x}\right )} \end{align*}
✓ Maple. Time used: 0.623 (sec). Leaf size: 110
ode:=diff(y(x),x) = (-b*y(x)*a+b^2+a*b+b^2*x-b*a*x^(1/2)-a^2)/a/(-a*y(x)+b+a+b*x-a*x^(1/2));
dsolve(ode,y(x), singsol=all);
\[
y = \frac {\operatorname {RootOf}\left (-x^{{3}/{2}} a b +b^{2} x^{2}-a^{2} \sqrt {x}-b a \sqrt {x}-2 a^{2} x +2 a x b +2 x \,b^{2}+a^{2}+2 a b +b^{2}+{\mathrm e}^{\operatorname {RootOf}\left (-4 \,{\mathrm e}^{\textit {\_Z}}+9 \operatorname {sech}\left (-\frac {3 \textit {\_Z}}{2}+\frac {c_{1}}{2}\right )^{2} a^{2} x \right )}+\left (a \sqrt {x}-2 b x -2 a -2 b \right ) \textit {\_Z} +\textit {\_Z}^{2}\right )}{a}
\]
✓ Mathematica. Time used: 60.101 (sec). Leaf size: 649
ode=D[y[x],x] == (-a^2 + a*b + b^2 - a*b*Sqrt[x] + b^2*x - a*b*y[x])/(a*(a + b - a*Sqrt[x] + b*x - a*y[x]));
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to -\frac {a \sqrt {x}-a-b x-b}{a}+\frac {1}{a^2 \text {Root}\left [\text {$\#$1}^6 \left (16 x^3+16 e^{12 c_1}\right )-\frac {24 \text {$\#$1}^4 x^2}{a^4}+\frac {8 \text {$\#$1}^3 x^{3/2}}{a^6}+\frac {9 \text {$\#$1}^2 x}{a^8}-\frac {6 \text {$\#$1} \sqrt {x}}{a^{10}}+\frac {1}{a^{12}}\&,1\right ]} \\
y(x)\to -\frac {a \sqrt {x}-a-b x-b}{a}+\frac {1}{a^2 \text {Root}\left [\text {$\#$1}^6 \left (16 x^3+16 e^{12 c_1}\right )-\frac {24 \text {$\#$1}^4 x^2}{a^4}+\frac {8 \text {$\#$1}^3 x^{3/2}}{a^6}+\frac {9 \text {$\#$1}^2 x}{a^8}-\frac {6 \text {$\#$1} \sqrt {x}}{a^{10}}+\frac {1}{a^{12}}\&,2\right ]} \\
y(x)\to -\frac {a \sqrt {x}-a-b x-b}{a}+\frac {1}{a^2 \text {Root}\left [\text {$\#$1}^6 \left (16 x^3+16 e^{12 c_1}\right )-\frac {24 \text {$\#$1}^4 x^2}{a^4}+\frac {8 \text {$\#$1}^3 x^{3/2}}{a^6}+\frac {9 \text {$\#$1}^2 x}{a^8}-\frac {6 \text {$\#$1} \sqrt {x}}{a^{10}}+\frac {1}{a^{12}}\&,3\right ]} \\
y(x)\to -\frac {a \sqrt {x}-a-b x-b}{a}+\frac {1}{a^2 \text {Root}\left [\text {$\#$1}^6 \left (16 x^3+16 e^{12 c_1}\right )-\frac {24 \text {$\#$1}^4 x^2}{a^4}+\frac {8 \text {$\#$1}^3 x^{3/2}}{a^6}+\frac {9 \text {$\#$1}^2 x}{a^8}-\frac {6 \text {$\#$1} \sqrt {x}}{a^{10}}+\frac {1}{a^{12}}\&,4\right ]} \\
y(x)\to -\frac {a \sqrt {x}-a-b x-b}{a}+\frac {1}{a^2 \text {Root}\left [\text {$\#$1}^6 \left (16 x^3+16 e^{12 c_1}\right )-\frac {24 \text {$\#$1}^4 x^2}{a^4}+\frac {8 \text {$\#$1}^3 x^{3/2}}{a^6}+\frac {9 \text {$\#$1}^2 x}{a^8}-\frac {6 \text {$\#$1} \sqrt {x}}{a^{10}}+\frac {1}{a^{12}}\&,5\right ]} \\
y(x)\to -\frac {a \sqrt {x}-a-b x-b}{a}+\frac {1}{a^2 \text {Root}\left [\text {$\#$1}^6 \left (16 x^3+16 e^{12 c_1}\right )-\frac {24 \text {$\#$1}^4 x^2}{a^4}+\frac {8 \text {$\#$1}^3 x^{3/2}}{a^6}+\frac {9 \text {$\#$1}^2 x}{a^8}-\frac {6 \text {$\#$1} \sqrt {x}}{a^{10}}+\frac {1}{a^{12}}\&,6\right ]} \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
y = Function("y")
ode = Eq(Derivative(y(x), x) - (-a**2 - a*b*sqrt(x) - a*b*y(x) + a*b + b**2*x + b**2)/(a*(-a*sqrt(x) - a*y(x) + a + b*x + b)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out