Internal problem ID [9675]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1341.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime }+\frac {\left (2 a x +b \right ) y^{\prime }}{x \left (a x +b \right )}+\frac {\left (a v x -b \right ) y}{\left (a x +b \right ) x^{2}}=A x} \]
✓ Solution by Maple
Time used: 0.157 (sec). Leaf size: 195
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)
\[ y \left (x \right ) = \operatorname {hypergeom}\left (\left [-\frac {1}{2}-\frac {\sqrt {1-4 v}}{2}, \frac {3}{2}-\frac {\sqrt {1-4 v}}{2}\right ], \left [1-\sqrt {1-4 v}\right ], -\frac {b}{x a}\right ) x^{-\frac {1}{2}+\frac {\sqrt {1-4 v}}{2}} c_{2} +\operatorname {hypergeom}\left (\left [-\frac {1}{2}+\frac {\sqrt {1-4 v}}{2}, \frac {3}{2}+\frac {\sqrt {1-4 v}}{2}\right ], \left [1+\sqrt {1-4 v}\right ], -\frac {b}{x a}\right ) x^{-\frac {1}{2}-\frac {\sqrt {1-4 v}}{2}} c_{1} +\frac {\left (a x \left (a x +b \right ) v^{2}+\left (8 a^{2} x^{2}+6 a b x -3 b^{2}\right ) v +12 a^{2} x^{2}+8 a b x -12 b^{2}\right ) A x}{a^{2} \left (v +6\right ) \left (2+v \right ) \left (v +12\right )} \]
✓ Solution by Mathematica
Time used: 71.383 (sec). Leaf size: 725
DSolve[y''[x] == A*x - ((-b + a*v*x)*y[x])/(x^2*(b + a*x)) - ((b + 2*a*x)*y'[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 ) \]