60.3.420 problem 1426
Internal
problem
ID
[11430]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
2,
linear
second
order
Problem
number
:
1426
Date
solved
:
Tuesday, January 28, 2025 at 06:05:37 PM
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*}
✓ Solution by Maple
Time used: 1.182 (sec). Leaf size: 558
dsolve(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,y(x), singsol=all)
\[
y = \frac {\left (\frac {\cos \left (x \right )}{2}-\frac {1}{2}\right )^{\frac {4 a -6+\sqrt {16 a^{4}+\left (16 b -72\right ) a^{2}-48 a b +4 \left (\frac {9}{2}+b \right )^{2}}}{-12+8 a}} \left (\operatorname {hypergeom}\left (\left [\frac {8 a^{2}-\sqrt {16 a^{4}+\left (-16 b -72\right ) a^{2}+48 a b +4 \left (b -\frac {9}{2}\right )^{2}}+\sqrt {16 a^{4}+\left (16 b -72\right ) a^{2}-48 a b +4 \left (\frac {9}{2}+b \right )^{2}}-8 a -6}{-12+8 a}, \frac {-8 a^{2}-\sqrt {16 a^{4}+\left (-16 b -72\right ) a^{2}+48 a b +4 \left (b -\frac {9}{2}\right )^{2}}+\sqrt {16 a^{4}+\left (16 b -72\right ) a^{2}-48 a b +4 \left (\frac {9}{2}+b \right )^{2}}+16 a -6}{-12+8 a}\right ], \left [\frac {4 a -6-\sqrt {16 a^{4}+\left (-16 b -72\right ) a^{2}+48 a b +4 \left (b -\frac {9}{2}\right )^{2}}}{4 a -6}\right ], \frac {\cos \left (x \right )}{2}+\frac {1}{2}\right ) \cos \left (\frac {x}{2}\right )^{\frac {4 a -6-\sqrt {16 a^{4}+\left (-16 b -72\right ) a^{2}+48 a b +4 \left (b -\frac {9}{2}\right )^{2}}}{4 a -6}} c_{1} +\cos \left (\frac {x}{2}\right )^{\frac {4 a -6+\sqrt {16 a^{4}+\left (-16 b -72\right ) a^{2}+48 a b +4 \left (b -\frac {9}{2}\right )^{2}}}{4 a -6}} \operatorname {hypergeom}\left (\left [\frac {8 a^{2}+\sqrt {16 a^{4}+\left (-16 b -72\right ) a^{2}+48 a b +4 \left (b -\frac {9}{2}\right )^{2}}+\sqrt {16 a^{4}+\left (16 b -72\right ) a^{2}-48 a b +4 \left (\frac {9}{2}+b \right )^{2}}-8 a -6}{-12+8 a}, \frac {-8 a^{2}+\sqrt {16 a^{4}+\left (-16 b -72\right ) a^{2}+48 a b +4 \left (b -\frac {9}{2}\right )^{2}}+\sqrt {16 a^{4}+\left (16 b -72\right ) a^{2}-48 a b +4 \left (\frac {9}{2}+b \right )^{2}}+16 a -6}{-12+8 a}\right ], \left [\frac {4 a -6+\sqrt {16 a^{4}+\left (-16 b -72\right ) a^{2}+48 a b +4 \left (b -\frac {9}{2}\right )^{2}}}{4 a -6}\right ], \frac {\cos \left (x \right )}{2}+\frac {1}{2}\right ) c_{2} \right )}{\sqrt {\sin \left (x \right )}}
\]
✓ Solution by Mathematica
Time used: 5.787 (sec). Leaf size: 1281
DSolve[(-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,y[x],x,IncludeSingularSolutions -> True]
\[
y(x)\to \frac {(-1)^{\frac {-4 a^2-9}{(3-2 a)^2}} 2^{-\frac {\sqrt {(3-2 a)^2 \left (16 a^4+8 (2 b-9) a^2-48 b a+(2 b+9)^2\right )}}{2 (3-2 a)^2}} (\cos (x)-1)^{-\frac {-8 a^2+24 a+\sqrt {(3-2 a)^2 \left (16 a^4+8 (2 b-9) a^2-48 b a+(2 b+9)^2\right )}-18}{4 (3-2 a)^2}} (\cos (x)+1)^{\frac {1}{4} \left (\sqrt {\frac {16 a^4-8 (2 b+9) a^2+48 b a+(9-2 b)^2}{(3-2 a)^2}}+2\right )} \left ((-1)^{\frac {4 a^2+9}{(3-2 a)^2}} 2^{\frac {\sqrt {(3-2 a)^2 \left (16 a^4+8 (2 b-9) a^2-48 b a+(2 b+9)^2\right )}}{2 (3-2 a)^2}} c_1 \operatorname {Hypergeometric2F1}\left (\frac {16 a^3+4 \left (\sqrt {\frac {16 a^4-8 (2 b+9) a^2+48 b a+(9-2 b)^2}{(3-2 a)^2}}-10\right ) a^2-12 \left (\sqrt {\frac {16 a^4-8 (2 b+9) a^2+48 b a+(9-2 b)^2}{(3-2 a)^2}}-1\right ) a+9 \sqrt {\frac {16 a^4-8 (2 b+9) a^2+48 b a+(9-2 b)^2}{(3-2 a)^2}}-\sqrt {(3-2 a)^2 \left (16 a^4+8 (2 b-9) a^2-48 b a+(2 b+9)^2\right )}+18}{4 (3-2 a)^2},-\frac {16 a^3-4 \left (\sqrt {\frac {16 a^4-8 (2 b+9) a^2+48 b a+(9-2 b)^2}{(3-2 a)^2}}+14\right ) a^2+12 \left (\sqrt {\frac {16 a^4-8 (2 b+9) a^2+48 b a+(9-2 b)^2}{(3-2 a)^2}}+5\right ) a-9 \sqrt {\frac {16 a^4-8 (2 b+9) a^2+48 b a+(9-2 b)^2}{(3-2 a)^2}}+\sqrt {(3-2 a)^2 \left (16 a^4+8 (2 b-9) a^2-48 b a+(2 b+9)^2\right )}-18}{4 (3-2 a)^2},-\frac {-8 a^2+24 a+\sqrt {(3-2 a)^2 \left (16 a^4+8 (2 b-9) a^2-48 b a+(2 b+9)^2\right )}-18}{2 (3-2 a)^2},\sin ^2\left (\frac {x}{2}\right )\right )-i^{\frac {24 a+\sqrt {(3-2 a)^2 \left (16 a^4+8 (2 b-9) a^2-48 b a+(2 b+9)^2\right )}}{(3-2 a)^2}} c_2 (1-\cos (x))^{\frac {\sqrt {(3-2 a)^2 \left (16 a^4+8 (2 b-9) a^2-48 b a+(2 b+9)^2\right )}}{2 (3-2 a)^2}} \operatorname {Hypergeometric2F1}\left (\frac {16 a^3+4 \left (\sqrt {\frac {16 a^4-8 (2 b+9) a^2+48 b a+(9-2 b)^2}{(3-2 a)^2}}-10\right ) a^2-12 \left (\sqrt {\frac {16 a^4-8 (2 b+9) a^2+48 b a+(9-2 b)^2}{(3-2 a)^2}}-1\right ) a+9 \sqrt {\frac {16 a^4-8 (2 b+9) a^2+48 b a+(9-2 b)^2}{(3-2 a)^2}}+\sqrt {(3-2 a)^2 \left (16 a^4+8 (2 b-9) a^2-48 b a+(2 b+9)^2\right )}+18}{4 (3-2 a)^2},\frac {-16 a^3+4 \left (\sqrt {\frac {16 a^4-8 (2 b+9) a^2+48 b a+(9-2 b)^2}{(3-2 a)^2}}+14\right ) a^2-12 \left (\sqrt {\frac {16 a^4-8 (2 b+9) a^2+48 b a+(9-2 b)^2}{(3-2 a)^2}}+5\right ) a+9 \sqrt {\frac {16 a^4-8 (2 b+9) a^2+48 b a+(9-2 b)^2}{(3-2 a)^2}}+\sqrt {(3-2 a)^2 \left (16 a^4+8 (2 b-9) a^2-48 b a+(2 b+9)^2\right )}+18}{4 (3-2 a)^2},\frac {8 a^2-24 a+\sqrt {(3-2 a)^2 \left (16 a^4+8 (2 b-9) a^2-48 b a+(2 b+9)^2\right )}+18}{2 (3-2 a)^2},\sin ^2\left (\frac {x}{2}\right )\right )\right )}{\sqrt [4]{-\sin ^2(x)}}
\]