61.32.20 problem 229
Internal
problem
ID
[12730]
Book
:
Handbook
of
exact
solutions
for
ordinary
differential
equations.
By
Polyanin
and
Zaitsev.
Second
edition
Section
:
Chapter
2,
Second-Order
Differential
Equations.
section
2.1.2-7
Problem
number
:
229
Date
solved
:
Tuesday, January 28, 2025 at 04:16:57 AM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} \left (a \,x^{2}+b \right )^{2} y^{\prime \prime }+\left (a \,x^{2}+b \right ) \left (c \,x^{2}+d \right ) y^{\prime }+2 \left (-a d +b c \right ) x y&=0 \end{align*}
✓ Solution by Maple
Time used: 0.980 (sec). Leaf size: 864
dsolve((a*x^2+b)^2*diff(y(x),x$2)+(a*x^2+b)*(c*x^2+d)*diff(y(x),x)+2*(b*c-a*d)*x*y(x)=0,y(x), singsol=all)
\[
y = \left (-a x +\sqrt {-a b}\right )^{\frac {2 a^{2} b +\sqrt {-a b \left (4 \sqrt {-a b}\, a^{2} d -4 \sqrt {-a b}\, a b c -4 a^{3} b +a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{4 a^{2} b}} \left (c_{1} \left (a x +\sqrt {-a b}\right )^{\frac {2 a^{2} b +\sqrt {4 a^{2} b \left (a d -b c \right ) \sqrt {-a b}+4 a^{4} b^{2}-a^{3} b \,d^{2}+2 b^{2} d c \,a^{2}-b^{3} c^{2} a}}{4 a^{2} b}} {\mathrm e}^{\frac {\sqrt {-a b}\, c}{2 a^{2}}-\frac {\arctan \left (\frac {\sqrt {a}\, x}{\sqrt {b}}\right ) d}{2 \sqrt {a}\, \sqrt {b}}+\frac {\sqrt {b}\, \arctan \left (\frac {\sqrt {a}\, x}{\sqrt {b}}\right ) c}{2 a^{{3}/{2}}}} \operatorname {HeunC}\left (\frac {2 \sqrt {-\frac {b}{a}}\, c}{a}, \frac {\sqrt {4 a^{2} b \left (a d -b c \right ) \sqrt {-a b}+4 a^{4} b^{2}-a^{3} b \,d^{2}+2 b^{2} d c \,a^{2}-b^{3} c^{2} a}}{2 a^{2} b}, \frac {\sqrt {-a b \left (4 \sqrt {-a b}\, a^{2} d -4 \sqrt {-a b}\, a b c -4 a^{3} b +a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{2 a^{2} b}, 0, \frac {1}{2}-\frac {d^{2}}{8 a b}-\frac {c d}{4 a^{2}}+\frac {3 b \,c^{2}}{8 a^{3}}, \frac {a x}{2 \sqrt {-a b}}+\frac {1}{2}\right )+c_{2} \left (a x +\sqrt {-a b}\right )^{-\frac {-2 a^{2} b +\sqrt {4 a^{2} b \left (a d -b c \right ) \sqrt {-a b}+4 a^{4} b^{2}-a^{3} b \,d^{2}+2 b^{2} d c \,a^{2}-b^{3} c^{2} a}}{4 a^{2} b}} \operatorname {HeunC}\left (\frac {2 \sqrt {-\frac {b}{a}}\, c}{a}, -\frac {\sqrt {4 a^{2} b \left (a d -b c \right ) \sqrt {-a b}+4 a^{4} b^{2}-a^{3} b \,d^{2}+2 b^{2} d c \,a^{2}-b^{3} c^{2} a}}{2 a^{2} b}, \frac {\sqrt {-a b \left (4 \sqrt {-a b}\, a^{2} d -4 \sqrt {-a b}\, a b c -4 a^{3} b +a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{2 a^{2} b}, 0, \frac {1}{2}-\frac {d^{2}}{8 a b}-\frac {c d}{4 a^{2}}+\frac {3 b \,c^{2}}{8 a^{3}}, \frac {a x}{2 \sqrt {-a b}}+\frac {1}{2}\right ) {\mathrm e}^{\frac {i \pi \sqrt {4 a^{2} b \left (a d -b c \right ) \sqrt {-a b}+4 a^{4} b^{2}-a^{3} b \,d^{2}+2 b^{2} d c \,a^{2}-b^{3} c^{2} a}-i \pi \sqrt {-a b \left (4 \sqrt {-a b}\, a^{2} d -4 \sqrt {-a b}\, a b c -4 a^{3} b +a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-4 \left (a^{2} \left (\frac {d}{\sqrt {b}\, \sqrt {a}}-\frac {\sqrt {b}\, c}{a^{{3}/{2}}}\right ) \arctan \left (\frac {\sqrt {a}\, x}{\sqrt {b}}\right )-\sqrt {-a b}\, c \right ) b}{8 a^{2} b}}\right )
\]
✓ Solution by Mathematica
Time used: 0.401 (sec). Leaf size: 116
DSolve[(a*x^2+b)^2*D[y[x],{x,2}]+(a*x^2+b)*(c*x^2+d)*D[y[x],x]+2*(b*c-a*d)*x*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \exp \left (\int _1^x-\frac {c K[1]^2+d}{a K[1]^2+b}dK[1]\right ) \left (\int _1^x\exp \left (-\int _1^{K[2]}-\frac {c K[1]^2+d}{a K[1]^2+b}dK[1]\right ) c_1dK[2]+c_2\right ) \\
y(x)\to c_2 \exp \left (\int _1^x-\frac {c K[1]^2+d}{a K[1]^2+b}dK[1]\right ) \\
\end{align*}