60.3.333 problem 1339

Internal problem ID [11343]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1339
Date solved : Tuesday, January 28, 2025 at 06:02:29 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }&=-\frac {\left (a \left (b +2\right ) x^{2}+\left (c -d +1\right ) x \right ) y^{\prime }}{\left (a x +1\right ) x^{2}}-\frac {\left (a b x -c d \right ) y}{\left (a x +1\right ) x^{2}} \end{align*}

✓ Solution by Maple

Time used: 0.267 (sec). Leaf size: 76

dsolve(diff(diff(y(x),x),x) = -(a*(b+2)*x^2+(c-d+1)*x)/(a*x+1)/x^2*diff(y(x),x)-(a*b*x-c*d)/(a*x+1)/x^2*y(x),y(x), singsol=all)
 

\[ y = \left (\operatorname {hypergeom}\left (\left [-d , 1-b -d \right ], \left [1-d -c \right ], -a x \right ) x^{-c} c_{2} +\operatorname {hypergeom}\left (\left [c , 1+c -b \right ], \left [1+d +c \right ], -a x \right ) x^{d} c_{1} \right ) \left (a x +1\right )^{-b +c -d} \]

✓ Solution by Mathematica

Time used: 0.279 (sec). Leaf size: 66

DSolve[D[y[x],{x,2}] == -(((-(c*d) + a*b*x)*y[x])/(x^2*(1 + a*x))) - (((1 + c - d)*x + a*(2 + b)*x^2)*D[y[x],x])/(x^2*(1 + a*x)),y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to c_1 a^{-c} x^{-c} \operatorname {Hypergeometric2F1}(1-c,b-c,-c-d+1,-a x)+c_2 a^d x^d \operatorname {Hypergeometric2F1}(d+1,b+d,c+d+1,-a x) \]