23.1.517 problem 507
Internal
problem
ID
[5124]
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
:
507
Date
solved
:
Tuesday, September 30, 2025 at 11:43:00 AM
CAS
classification
:
[[_homogeneous, `class C`], _rational, [_Abel, `2nd type`, `class A`]]
\begin{align*} \left (a_{2} +b_{2} x +c_{2} y\right ) y^{\prime }&=a_{1} +b_{1} x +c_{1} y \end{align*}
✓ Maple. Time used: 0.116 (sec). Leaf size: 232
ode:=(a__2+b__2*x+c__2*y(x))*diff(y(x),x) = a__1+b__1*x+c__1*y(x);
dsolve(ode,y(x), singsol=all);
\[
y = \frac {\left (\left (-b_{2} x -a_{2} \right ) c_{1} +c_{2} \left (b_{1} x +a_{1} \right )\right ) \sqrt {-4 b_{1} c_{2} -b_{2}^{2}+2 b_{2} c_{1} -c_{1}^{2}}\, \tan \left (\operatorname {RootOf}\left (\sqrt {-4 b_{1} c_{2} -b_{2}^{2}+2 b_{2} c_{1} -c_{1}^{2}}\, \ln \left (-\frac {\left (b_{1} c_{2} x -b_{2} c_{1} x +a_{1} c_{2} -a_{2} c_{1} \right )^{2} \left (4 b_{1} c_{2} +b_{2}^{2}-2 b_{2} c_{1} +c_{1}^{2}\right ) \sec \left (\textit {\_Z} \right )^{2}}{c_{2}}\right )-2 \sqrt {-4 b_{1} c_{2} -b_{2}^{2}+2 b_{2} c_{1} -c_{1}^{2}}\, \ln \left (2\right )+2 c_3 \sqrt {-4 b_{1} c_{2} -b_{2}^{2}+2 b_{2} c_{1} -c_{1}^{2}}+2 \textit {\_Z} b_{2} +2 \textit {\_Z} c_{1} \right )\right )+\left (-b_{2} x -a_{2} \right ) c_{1}^{2}+\left (c_{2} \left (b_{1} x +a_{1} \right )+b_{2} \left (b_{2} x +a_{2} \right )\right ) c_{1} -c_{2} \left (\left (b_{1} x -a_{1} \right ) b_{2} +2 a_{2} b_{1} \right )}{2 c_{2} \left (b_{1} c_{2} -b_{2} c_{1} \right )}
\]
✓ Mathematica. Time used: 0.872 (sec). Leaf size: 273
ode=(a2+b2*x+c2*y[x])*D[y[x],x]==(a1+b1*x+c1*y[x]);
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [-\frac {(\text {b2}+\text {c1})^2 \left (\sqrt {\frac {4 (\text {b2} \text {c1}-\text {b1} \text {c2})}{(\text {b2}+\text {c1})^2}-1} \log \left (\frac {(\text {a2}+\text {b2} x+\text {c2} y(x)) \left ((\text {b2} \text {c1}-\text {b1} \text {c2}) (\text {a2}+\text {b2} x+\text {c2} y(x))-\frac {(-\text {a1} \text {c2}+\text {a2} \text {c1}-\text {b1} \text {c2} x+\text {b2} \text {c1} x) \left (\text {a1} \text {c2}+\text {a2} \text {b2}+\text {b1} \text {c2} x+\text {b2}^2 x+\text {c2} (\text {b2}+\text {c1}) y(x)\right )}{\text {a2}+\text {b2} x+\text {c2} y(x)}\right )}{(-\text {a1} \text {c2}+\text {a2} \text {c1}-\text {b1} \text {c2} x+\text {b2} \text {c1} x)^2}\right )-2 \arctan \left (\frac {\frac {2 (-\text {a1} \text {c2}+\text {a2} \text {c1}-\text {b1} \text {c2} x+\text {b2} \text {c1} x)}{\text {a2}+\text {b2} x+\text {c2} y(x)}-\text {b2}-\text {c1}}{(\text {b2}+\text {c1}) \sqrt {\frac {4 (\text {b2} \text {c1}-\text {b1} \text {c2})}{(\text {b2}+\text {c1})^2}-1}}\right )\right )}{2 (\text {b2} \text {c1}-\text {b1} \text {c2}) \sqrt {\frac {4 (\text {b2} \text {c1}-\text {b1} \text {c2})}{(\text {b2}+\text {c1})^2}-1}}=\frac {(\text {b2}+\text {c1})^2 \log (-\text {a1} \text {c2}+\text {a2} \text {c1}-\text {b1} \text {c2} x+\text {b2} \text {c1} x)}{\text {b2} \text {c1}-\text {b1} \text {c2}}+c_1,y(x)\right ]
\]
✗ Sympy
from sympy import *
x = symbols("x")
a2 = symbols("a2")
b2 = symbols("b2")
c2 = symbols("c2")
a1 = symbols("a1")
b1 = symbols("b1")
c1 = symbols("c1")
y = Function("y")
ode = Eq(-a1 - b1*x - c1*y(x) + (a2 + b2*x + c2*y(x))*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out