29.13.23 problem 377
Internal
problem
ID
[4977]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
13
Problem
number
:
377
Date
solved
:
Tuesday, March 04, 2025 at 07:39:47 PM
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: 0.007 (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);
\[
y \left (x \right ) = \frac {2 c \left (c_{1} \left (i \sqrt {\frac {-4 a c +b^{2}-4 A}{c^{2}}}\, c \sqrt {4 a c -b^{2}}-\sqrt {-4 a c +b^{2}}\, \left (2 c x +b \right )\right ) {\left (\frac {-b +i \sqrt {4 a c -b^{2}}-2 c x}{b +i \sqrt {4 a c -b^{2}}+2 c x}\right )}^{-\frac {c \sqrt {\frac {-4 a c +b^{2}-4 A}{c^{2}}}}{2 \sqrt {-4 a c +b^{2}}}}-\left (i \sqrt {\frac {-4 a c +b^{2}-4 A}{c^{2}}}\, c \sqrt {4 a c -b^{2}}+\sqrt {-4 a c +b^{2}}\, \left (2 c x +b \right )\right ) {\left (\frac {-b +i \sqrt {4 a c -b^{2}}-2 c x}{b +i \sqrt {4 a c -b^{2}}+2 c x}\right )}^{\frac {c \sqrt {\frac {-4 a c +b^{2}-4 A}{c^{2}}}}{2 \sqrt {-4 a c +b^{2}}}}\right )}{\sqrt {-4 a c +b^{2}}\, \left (b +i \sqrt {4 a c -b^{2}}+2 c x \right ) \left (-b +i \sqrt {4 a c -b^{2}}-2 c x \right ) \left (c_{1} {\left (\frac {-b +i \sqrt {4 a c -b^{2}}-2 c x}{b +i \sqrt {4 a c -b^{2}}+2 c x}\right )}^{-\frac {c \sqrt {\frac {-4 a c +b^{2}-4 A}{c^{2}}}}{2 \sqrt {-4 a c +b^{2}}}}+{\left (\frac {-b +i \sqrt {4 a c -b^{2}}-2 c x}{b +i \sqrt {4 a c -b^{2}}+2 c x}\right )}^{\frac {c \sqrt {\frac {-4 a c +b^{2}-4 A}{c^{2}}}}{2 \sqrt {-4 a c +b^{2}}}}\right )}
\]
✓ Mathematica. Time used: 3.217 (sec). Leaf size: 743
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 {b^2 c_1 \left (-\exp \left (\frac {2 \sqrt {4 a c-b^2} \sqrt {1-\frac {4 A}{b^2-4 a c}} \arctan \left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )}{\sqrt {b^2-4 a c}}\right )\right )+b c_1 \sqrt {b^2-4 a c} \sqrt {1-\frac {4 A}{b^2-4 a c}} \exp \left (\frac {2 \sqrt {4 a c-b^2} \sqrt {1-\frac {4 A}{b^2-4 a c}} \arctan \left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )}{\sqrt {b^2-4 a c}}\right )+4 A c_1 \exp \left (\frac {2 \sqrt {4 a c-b^2} \sqrt {1-\frac {4 A}{b^2-4 a c}} \arctan \left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )}{\sqrt {b^2-4 a c}}\right )+4 a c c_1 \exp \left (\frac {2 \sqrt {4 a c-b^2} \sqrt {1-\frac {4 A}{b^2-4 a c}} \arctan \left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )}{\sqrt {b^2-4 a c}}\right )+2 c c_1 x \sqrt {b^2-4 a c} \sqrt {1-\frac {4 A}{b^2-4 a c}} \exp \left (\frac {2 \sqrt {4 a c-b^2} \sqrt {1-\frac {4 A}{b^2-4 a c}} \arctan \left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )}{\sqrt {b^2-4 a c}}\right )+\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)) \left (1+c_1 \sqrt {b^2-4 a c} \sqrt {1-\frac {4 A}{b^2-4 a c}} \exp \left (\frac {2 \sqrt {4 a c-b^2} \sqrt {1-\frac {4 A}{b^2-4 a c}} \arctan \left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )}{\sqrt {b^2-4 a c}}\right )\right )} \\
y(x)\to \frac {2 c x \sqrt {b^2-4 a c} \sqrt {1-\frac {4 A}{b^2-4 a c}}+b \sqrt {b^2-4 a c} \sqrt {1-\frac {4 A}{b^2-4 a c}}+4 a c+4 A-b^2}{2 \sqrt {b^2-4 a c} \sqrt {1-\frac {4 A}{b^2-4 a c}} (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