54.2.149 problem 726
Internal
problem
ID
[12023]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
Additional
non-linear
first
order
Problem
number
:
726
Date
solved
:
Tuesday, September 30, 2025 at 11:57:24 PM
CAS
classification
:
[[_1st_order, _with_linear_symmetries], _rational, [_Abel, `2nd type`, `class A`]]
\begin{align*} y^{\prime }&=-\frac {b y a -b c +b^{2} x +b a \sqrt {x}-a^{2}}{a \left (a y-c +b x +a \sqrt {x}\right )} \end{align*}
✓ Maple. Time used: 0.126 (sec). Leaf size: 87
ode:=diff(y(x),x) = -(b*y(x)*a-b*c+b^2*x+b*a*x^(1/2)-a^2)/a/(a*y(x)-c+b*x+a*x^(1/2));
dsolve(ode,y(x), singsol=all);
\[
y = \frac {\operatorname {RootOf}\left (x^{{3}/{2}} a b +b^{2} x^{2}-\sqrt {x}\, a c -2 a^{2} x -2 b c x +c^{2}-{\mathrm e}^{\operatorname {RootOf}\left (4 \,{\mathrm e}^{\textit {\_Z}}+9 a^{2} x \operatorname {sech}\left (-\frac {3 \textit {\_Z}}{2}+\frac {c_1}{2}\right )^{2}\right )}+\left (a \sqrt {x}+2 b x -2 c \right ) \textit {\_Z} +\textit {\_Z}^{2}\right )}{a}
\]
✓ Mathematica. Time used: 60.05 (sec). Leaf size: 625
ode=D[y[x],x] == (a^2 + b*c - a*b*Sqrt[x] - b^2*x - a*b*y[x])/(a*(-c + 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}+b x-c}{a}+\frac {1}{a^2 \text {Root}\left [\text {$\#$1}^6 \left (16 x^3+16 e^{12 c_1}\right )-\frac {24 \text {$\#$1}^4 x^2}{a^4}+\frac {8 \text {$\#$1}^3 x^{3/2}}{a^6}+\frac {9 \text {$\#$1}^2 x}{a^8}-\frac {6 \text {$\#$1} \sqrt {x}}{a^{10}}+\frac {1}{a^{12}}\&,1\right ]}\\ y(x)&\to -\frac {a \sqrt {x}+b x-c}{a}+\frac {1}{a^2 \text {Root}\left [\text {$\#$1}^6 \left (16 x^3+16 e^{12 c_1}\right )-\frac {24 \text {$\#$1}^4 x^2}{a^4}+\frac {8 \text {$\#$1}^3 x^{3/2}}{a^6}+\frac {9 \text {$\#$1}^2 x}{a^8}-\frac {6 \text {$\#$1} \sqrt {x}}{a^{10}}+\frac {1}{a^{12}}\&,2\right ]}\\ y(x)&\to -\frac {a \sqrt {x}+b x-c}{a}+\frac {1}{a^2 \text {Root}\left [\text {$\#$1}^6 \left (16 x^3+16 e^{12 c_1}\right )-\frac {24 \text {$\#$1}^4 x^2}{a^4}+\frac {8 \text {$\#$1}^3 x^{3/2}}{a^6}+\frac {9 \text {$\#$1}^2 x}{a^8}-\frac {6 \text {$\#$1} \sqrt {x}}{a^{10}}+\frac {1}{a^{12}}\&,3\right ]}\\ y(x)&\to -\frac {a \sqrt {x}+b x-c}{a}+\frac {1}{a^2 \text {Root}\left [\text {$\#$1}^6 \left (16 x^3+16 e^{12 c_1}\right )-\frac {24 \text {$\#$1}^4 x^2}{a^4}+\frac {8 \text {$\#$1}^3 x^{3/2}}{a^6}+\frac {9 \text {$\#$1}^2 x}{a^8}-\frac {6 \text {$\#$1} \sqrt {x}}{a^{10}}+\frac {1}{a^{12}}\&,4\right ]}\\ y(x)&\to -\frac {a \sqrt {x}+b x-c}{a}+\frac {1}{a^2 \text {Root}\left [\text {$\#$1}^6 \left (16 x^3+16 e^{12 c_1}\right )-\frac {24 \text {$\#$1}^4 x^2}{a^4}+\frac {8 \text {$\#$1}^3 x^{3/2}}{a^6}+\frac {9 \text {$\#$1}^2 x}{a^8}-\frac {6 \text {$\#$1} \sqrt {x}}{a^{10}}+\frac {1}{a^{12}}\&,5\right ]}\\ y(x)&\to -\frac {a \sqrt {x}+b x-c}{a}+\frac {1}{a^2 \text {Root}\left [\text {$\#$1}^6 \left (16 x^3+16 e^{12 c_1}\right )-\frac {24 \text {$\#$1}^4 x^2}{a^4}+\frac {8 \text {$\#$1}^3 x^{3/2}}{a^6}+\frac {9 \text {$\#$1}^2 x}{a^8}-\frac {6 \text {$\#$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")
c = symbols("c")
y = Function("y")
ode = Eq(Derivative(y(x), x) + (-a**2 + a*b*sqrt(x) + a*b*y(x) + b**2*x - b*c)/(a*(a*sqrt(x) + a*y(x) + b*x - c)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out