54.3.409 problem 1426
Internal
problem
ID
[12704]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
2,
linear
second
order
Problem
number
:
1426
Date
solved
:
Friday, October 03, 2025 at 03:46:28 AM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} \sin \left (x \right )^{2} y^{\prime \prime }-\left (a^{2} \cos \left (x \right )^{2}+b \cos \left (x \right )+\frac {b^{2}}{\left (2 a -3\right )^{2}}+3 a +2\right ) y&=0 \end{align*}
✓ Maple. Time used: 0.440 (sec). Leaf size: 551
ode:=sin(x)^2*diff(diff(y(x),x),x)-(a^2*cos(x)^2+b*cos(x)+b^2/(2*a-3)^2+3*a+2)*y(x) = 0;
dsolve(ode,y(x), singsol=all);
\[
y = \frac {\cos \left (\frac {x}{2}\right ) \left (\frac {\cos \left (x \right )}{2}-\frac {1}{2}\right )^{\frac {4 a -6+\sqrt {16 a^{4}+\left (16 b -72\right ) a^{2}-48 a b +4 \left (\frac {9}{2}+b \right )^{2}}}{8 a -12}} \left (\cos \left (\frac {x}{2}\right )^{-\frac {\sqrt {16 a^{4}+\left (-16 b -72\right ) a^{2}+48 a b +4 \left (b -\frac {9}{2}\right )^{2}}}{4 a -6}} \operatorname {hypergeom}\left (\left [\frac {8 a^{2}+\sqrt {16 a^{4}+\left (16 b -72\right ) a^{2}-48 a b +4 \left (\frac {9}{2}+b \right )^{2}}-\sqrt {16 a^{4}+\left (-16 b -72\right ) a^{2}+48 a b +4 \left (b -\frac {9}{2}\right )^{2}}-8 a -6}{8 a -12}, \frac {-8 a^{2}+\sqrt {16 a^{4}+\left (16 b -72\right ) a^{2}-48 a b +4 \left (\frac {9}{2}+b \right )^{2}}-\sqrt {16 a^{4}+\left (-16 b -72\right ) a^{2}+48 a b +4 \left (b -\frac {9}{2}\right )^{2}}+16 a -6}{8 a -12}\right ], \left [\frac {4 a -6-\sqrt {16 a^{4}+\left (-16 b -72\right ) a^{2}+48 a b +4 \left (b -\frac {9}{2}\right )^{2}}}{4 a -6}\right ], \frac {\cos \left (x \right )}{2}+\frac {1}{2}\right ) c_1 +\cos \left (\frac {x}{2}\right )^{\frac {\sqrt {16 a^{4}+\left (-16 b -72\right ) a^{2}+48 a b +4 \left (b -\frac {9}{2}\right )^{2}}}{4 a -6}} \operatorname {hypergeom}\left (\left [\frac {8 a^{2}+\sqrt {16 a^{4}+\left (16 b -72\right ) a^{2}-48 a b +4 \left (\frac {9}{2}+b \right )^{2}}+\sqrt {16 a^{4}+\left (-16 b -72\right ) a^{2}+48 a b +4 \left (b -\frac {9}{2}\right )^{2}}-8 a -6}{8 a -12}, \frac {-8 a^{2}+\sqrt {16 a^{4}+\left (16 b -72\right ) a^{2}-48 a b +4 \left (\frac {9}{2}+b \right )^{2}}+\sqrt {16 a^{4}+\left (-16 b -72\right ) a^{2}+48 a b +4 \left (b -\frac {9}{2}\right )^{2}}+16 a -6}{8 a -12}\right ], \left [\frac {4 a -6+\sqrt {16 a^{4}+\left (-16 b -72\right ) a^{2}+48 a b +4 \left (b -\frac {9}{2}\right )^{2}}}{4 a -6}\right ], \frac {\cos \left (x \right )}{2}+\frac {1}{2}\right ) c_2 \right )}{\sqrt {\sin \left (x \right )}}
\]
✓ Mathematica. Time used: 3.425 (sec). Leaf size: 1281
ode=(-2 - 3*a - b^2/(-3 + 2*a)^2 - b*Cos[x] - a^2*Cos[x]^2)*y[x] + Sin[x]^2*D[y[x],{x,2}] == 0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} \text {Solution too large to show}\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
y = Function("y")
ode = Eq((-a**2*cos(x)**2 - 3*a - b**2/(2*a - 3)**2 - b*cos(x) - 2)*y(x) + sin(x)**2*Derivative(y(x), (x, 2)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : solve: Cannot solve (-a**2*cos(x)**2 - 3*a - b**2/(2*a - 3)**2 - b*cos(x) - 2)*y(x) + sin(x)**2*Derivative(y(x), (x, 2))