61.2.20 problem 20
Internal
problem
ID
[12026]
Book
:
Handbook
of
exact
solutions
for
ordinary
differential
equations.
By
Polyanin
and
Zaitsev.
Second
edition
Section
:
Chapter
1,
section
1.2.
Riccati
Equation.
1.2.2.
Equations
Containing
Power
Functions
Problem
number
:
20
Date
solved
:
Monday, January 27, 2025 at 11:51:27 PM
CAS
classification
:
[_rational, _Riccati]
\begin{align*} \left (a \,x^{2}+b x +c \right )^{2} \left (y^{\prime }+y^{2}\right )+A&=0 \end{align*}
✓ Solution by Maple
Time used: 0.004 (sec). Leaf size: 481
dsolve((a*x^2+b*x+c)^2*(diff(y(x),x)+y(x)^2)+A=0,y(x), singsol=all)
\[
y = \frac {2 \left (c_{1} \left (i \sqrt {\frac {-4 a c +b^{2}-4 A}{a^{2}}}\, a \sqrt {4 a c -b^{2}}-2 \sqrt {-4 a c +b^{2}}\, \left (a x +\frac {b}{2}\right )\right ) {\left (\frac {-b +i \sqrt {4 a c -b^{2}}-2 a x}{i \sqrt {4 a c -b^{2}}+2 a x +b}\right )}^{-\frac {a \sqrt {\frac {-4 a c +b^{2}-4 A}{a^{2}}}}{2 \sqrt {-4 a c +b^{2}}}}-{\left (\frac {-b +i \sqrt {4 a c -b^{2}}-2 a x}{i \sqrt {4 a c -b^{2}}+2 a x +b}\right )}^{\frac {a \sqrt {\frac {-4 a c +b^{2}-4 A}{a^{2}}}}{2 \sqrt {-4 a c +b^{2}}}} \left (i \sqrt {\frac {-4 a c +b^{2}-4 A}{a^{2}}}\, a \sqrt {4 a c -b^{2}}+2 \sqrt {-4 a c +b^{2}}\, \left (a x +\frac {b}{2}\right )\right )\right ) a}{\sqrt {-4 a c +b^{2}}\, \left (i \sqrt {4 a c -b^{2}}+2 a x +b \right ) \left (-b +i \sqrt {4 a c -b^{2}}-2 a x \right ) \left (c_{1} {\left (\frac {-b +i \sqrt {4 a c -b^{2}}-2 a x}{i \sqrt {4 a c -b^{2}}+2 a x +b}\right )}^{-\frac {a \sqrt {\frac {-4 a c +b^{2}-4 A}{a^{2}}}}{2 \sqrt {-4 a c +b^{2}}}}+{\left (\frac {-b +i \sqrt {4 a c -b^{2}}-2 a x}{i \sqrt {4 a c -b^{2}}+2 a x +b}\right )}^{\frac {a \sqrt {\frac {-4 a c +b^{2}-4 A}{a^{2}}}}{2 \sqrt {-4 a c +b^{2}}}}\right )}
\]
✓ Solution by Mathematica
Time used: 1.988 (sec). Leaf size: 397
DSolve[(a*x^2+b*x+c)^2*(D[y[x],x]+y[x]^2)+A==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {2 (x (a x+b)+c) \exp \left (-2 \int _1^x\frac {b+2 a K[1]-\sqrt {b^2-4 a c} \sqrt {1-\frac {4 A}{b^2-4 a c}}}{2 (c+K[1] (b+a K[1]))}dK[1]\right )+\left (-\sqrt {b^2-4 a c} \sqrt {1-\frac {4 A}{b^2-4 a c}}+2 a x+b\right ) \int _1^x\exp \left (-2 \int _1^{K[2]}\frac {b+2 a K[1]-\sqrt {b^2-4 a c} \sqrt {1-\frac {4 A}{b^2-4 a c}}}{2 (c+K[1] (b+a 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}}+2 a x+b\right )}{2 (x (a x+b)+c) \left (\int _1^x\exp \left (-2 \int _1^{K[2]}\frac {b+2 a K[1]-\sqrt {b^2-4 a c} \sqrt {1-\frac {4 A}{b^2-4 a c}}}{2 (c+K[1] (b+a 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}}+2 a x+b}{2 (x (a x+b)+c)} \\
\end{align*}