23.1.395 problem 380
Internal
problem
ID
[5002]
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
:
380
Date
solved
:
Tuesday, September 30, 2025 at 11:15:48 AM
CAS
classification
:
[_rational, _Riccati]
\begin{align*} \left (c \,x^{2}+b x +a \right )^{2} \left (y^{\prime }+y^{2}\right )+A&=0 \end{align*}
✓ Maple. Time used: 0.009 (sec). Leaf size: 478
ode:=(c*x^2+b*x+a)^2*(diff(y(x),x)+y(x)^2)+A = 0;
dsolve(ode,y(x), singsol=all);
\[
y = \frac {2 \left (\left (i \sqrt {\frac {-4 a c +b^{2}-4 A}{c^{2}}}\, c \sqrt {4 a c -b^{2}}-\sqrt {-4 a c +b^{2}}\, \left (2 c x +b \right )\right ) c_1 {\left (\frac {i \sqrt {4 a c -b^{2}}-2 c x -b}{2 c x +b +i \sqrt {4 a c -b^{2}}}\right )}^{-\frac {c \sqrt {\frac {-4 a c +b^{2}-4 A}{c^{2}}}}{2 \sqrt {-4 a c +b^{2}}}}-\left (i \sqrt {\frac {-4 a c +b^{2}-4 A}{c^{2}}}\, c \sqrt {4 a c -b^{2}}+\sqrt {-4 a c +b^{2}}\, \left (2 c x +b \right )\right ) {\left (\frac {i \sqrt {4 a c -b^{2}}-2 c x -b}{2 c x +b +i \sqrt {4 a c -b^{2}}}\right )}^{\frac {c \sqrt {\frac {-4 a c +b^{2}-4 A}{c^{2}}}}{2 \sqrt {-4 a c +b^{2}}}}\right ) c}{\sqrt {-4 a c +b^{2}}\, \left (2 c x +b +i \sqrt {4 a c -b^{2}}\right ) \left (i \sqrt {4 a c -b^{2}}-2 c x -b \right ) \left (c_1 {\left (\frac {i \sqrt {4 a c -b^{2}}-2 c x -b}{2 c x +b +i \sqrt {4 a c -b^{2}}}\right )}^{-\frac {c \sqrt {\frac {-4 a c +b^{2}-4 A}{c^{2}}}}{2 \sqrt {-4 a c +b^{2}}}}+{\left (\frac {i \sqrt {4 a c -b^{2}}-2 c x -b}{2 c x +b +i \sqrt {4 a c -b^{2}}}\right )}^{\frac {c \sqrt {\frac {-4 a c +b^{2}-4 A}{c^{2}}}}{2 \sqrt {-4 a c +b^{2}}}}\right )}
\]
✓ Mathematica. Time used: 1.542 (sec). Leaf size: 397
ode=(a+b*x+c*x^2)^2*(D[y[x],x]+y[x]^2)+A==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {2 (a+x (b+c x)) \exp \left (-2 \int _1^x\frac {b+2 c K[1]-\sqrt {b^2-4 a c} \sqrt {1-\frac {4 A}{b^2-4 a c}}}{2 (a+K[1] (b+c K[1]))}dK[1]\right )+\left (-\sqrt {b^2-4 a c} \sqrt {1-\frac {4 A}{b^2-4 a c}}+b+2 c x\right ) \int _1^x\exp \left (-2 \int _1^{K[2]}\frac {b+2 c K[1]-\sqrt {b^2-4 a c} \sqrt {1-\frac {4 A}{b^2-4 a c}}}{2 (a+K[1] (b+c K[1]))}dK[1]\right )dK[2]+c_1 \left (-\sqrt {b^2-4 a c} \sqrt {1-\frac {4 A}{b^2-4 a c}}+b+2 c x\right )}{2 (a+x (b+c x)) \left (\int _1^x\exp \left (-2 \int _1^{K[2]}\frac {b+2 c K[1]-\sqrt {b^2-4 a c} \sqrt {1-\frac {4 A}{b^2-4 a c}}}{2 (a+K[1] (b+c K[1]))}dK[1]\right )dK[2]+c_1\right )}\\ y(x)&\to \frac {-\sqrt {b^2-4 a c} \sqrt {1-\frac {4 A}{b^2-4 a c}}+b+2 c x}{2 (a+x (b+c x))} \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
A = symbols("A")
a = symbols("a")
b = symbols("b")
c = symbols("c")
y = Function("y")
ode = Eq(A + (y(x)**2 + Derivative(y(x), x))*(a + b*x + c*x**2)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE Derivative(y(x), x) - (-A - a**2*y(x)**2 - 2*a*b*x*y(x)**2 - 2*a*c*x**2*y(x)**2 - b**2*x**2*y(x)**2 - 2*b*c*x**3*y(x)**2 - c**2*x**4*y(x)**2)/(a**2 + 2*a*b*x + 2*a*c*x**2 + b**2*x**2 + 2*b*c*x**3 + c**2*x**4) cannot be solved by the factorable group method