60.3.335 problem 1341
Internal
problem
ID
[11345]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
2,
linear
second
order
Problem
number
:
1341
Date
solved
:
Tuesday, January 28, 2025 at 06:02:31 PM
CAS
classification
:
[[_2nd_order, _linear, _nonhomogeneous]]
\begin{align*} y^{\prime \prime }&=-\frac {\left (2 a x +b \right ) y^{\prime }}{x \left (a x +b \right )}-\frac {\left (a v x -b \right ) y}{\left (a x +b \right ) x^{2}}+A x \end{align*}
✓ Solution by Maple
Time used: 0.352 (sec). Leaf size: 194
dsolve(diff(diff(y(x),x),x) = -1/x*(2*a*x+b)/(a*x+b)*diff(y(x),x)-(a*v*x-b)/(a*x+b)/x^2*y(x)+A*x,y(x), singsol=all)
\[
y = \frac {x^{-\frac {\sqrt {1-4 v}}{2}} a^{2} c_{1} \left (v +6\right ) \left (2+v \right ) \left (v +12\right ) \operatorname {hypergeom}\left (\left [-\frac {1}{2}+\frac {\sqrt {1-4 v}}{2}, \frac {3}{2}+\frac {\sqrt {1-4 v}}{2}\right ], \left [1+\sqrt {1-4 v}\right ], -\frac {b}{a x}\right )-3 A \,b^{2} \left (v +4\right ) x^{{3}/{2}}+\left (2+v \right ) a \left (A b \left (v +4\right ) x^{{5}/{2}}+\left (A \,x^{{7}/{2}}+\left (v +12\right ) x^{\frac {\sqrt {1-4 v}}{2}} c_{2} \operatorname {hypergeom}\left (\left [-\frac {1}{2}-\frac {\sqrt {1-4 v}}{2}, \frac {3}{2}-\frac {\sqrt {1-4 v}}{2}\right ], \left [1-\sqrt {1-4 v}\right ], -\frac {b}{a x}\right )\right ) a \left (v +6\right )\right )}{\sqrt {x}\, a^{2} \left (v +6\right ) \left (2+v \right ) \left (v +12\right )}
\]
✓ Solution by Mathematica
Time used: 23.607 (sec). Leaf size: 725
DSolve[D[y[x],{x,2}] == A*x - ((-b + a*v*x)*y[x])/(x^2*(b + a*x)) - ((b + 2*a*x)*D[y[x],x])/(x*(b + a*x)),y[x],x,IncludeSingularSolutions -> True]
\[
y(x)\to \frac {a x \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (3-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+3\right ),3,-\frac {a x}{b}\right ) \left (\int _1^x-\frac {3 A b^2 K[1] G_{2,2}^{2,0}\left (-\frac {a K[1]}{b}| \begin {array}{c} \frac {1}{2} \left (1-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+1\right ) \\ -1,1 \\ \end {array} \right )}{a \left (\left (a (v+2) \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (5-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+5\right ),4,-\frac {a K[1]}{b}\right ) K[1]-3 b \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (3-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+3\right ),3,-\frac {a K[1]}{b}\right )\right ) G_{2,2}^{2,0}\left (-\frac {a K[1]}{b}| \begin {array}{c} \frac {1}{2} \left (1-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+1\right ) \\ -1,1 \\ \end {array} \right )+3 a \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (3-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+3\right ),3,-\frac {a K[1]}{b}\right ) K[1] G_{3,3}^{3,0}\left (-\frac {a K[1]}{b}| \begin {array}{c} -1,\frac {1}{2} \left (-\sqrt {1-4 v}-1\right ),\frac {1}{2} \left (\sqrt {1-4 v}-1\right ) \\ -2,0,0 \\ \end {array} \right )\right )}dK[1]+c_1\right )}{b}+G_{2,2}^{2,0}\left (-\frac {a x}{b}| \begin {array}{c} \frac {1}{2} \left (1-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+1\right ) \\ -1,1 \\ \end {array} \right ) \left (\int _1^x-\frac {3 A b \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (3-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+3\right ),3,-\frac {a K[2]}{b}\right ) K[2]^2}{\left (3 b \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (3-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+3\right ),3,-\frac {a K[2]}{b}\right )-a (v+2) \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (5-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+5\right ),4,-\frac {a K[2]}{b}\right ) K[2]\right ) G_{2,2}^{2,0}\left (-\frac {a K[2]}{b}| \begin {array}{c} \frac {1}{2} \left (1-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+1\right ) \\ -1,1 \\ \end {array} \right )-3 a \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (3-\sqrt {1-4 v}\right ),\frac {1}{2} \left (\sqrt {1-4 v}+3\right ),3,-\frac {a K[2]}{b}\right ) K[2] G_{3,3}^{3,0}\left (-\frac {a K[2]}{b}| \begin {array}{c} -1,\frac {1}{2} \left (-\sqrt {1-4 v}-1\right ),\frac {1}{2} \left (\sqrt {1-4 v}-1\right ) \\ -2,0,0 \\ \end {array} \right )}dK[2]+c_2\right )
\]