60.3.247 problem 1252

Internal problem ID [11257]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1252
Date solved : Tuesday, January 28, 2025 at 05:52:53 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x \left (1+x \right ) y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+c y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.230 (sec). Leaf size: 124

dsolve(x*(x+1)*diff(diff(y(x),x),x)+(a*x+b)*diff(y(x),x)+c*y(x)=0,y(x), singsol=all)
 

\[ y = c_{1} \operatorname {hypergeom}\left (\left [-\frac {1}{2}+\frac {\sqrt {a^{2}-2 a -4 c +1}}{2}+\frac {a}{2}, -\frac {1}{2}-\frac {\sqrt {a^{2}-2 a -4 c +1}}{2}+\frac {a}{2}\right ], \left [a -b \right ], x +1\right )+c_{2} \left (x +1\right )^{-a +b +1} \operatorname {hypergeom}\left (\left [\frac {1}{2}-\frac {\sqrt {a^{2}-2 a -4 c +1}}{2}+b -\frac {a}{2}, \frac {1}{2}+\frac {\sqrt {a^{2}-2 a -4 c +1}}{2}+b -\frac {a}{2}\right ], \left [2-a +b \right ], x +1\right ) \]

✓ Solution by Mathematica

Time used: 0.184 (sec). Leaf size: 131

DSolve[c*y[x] + (b + a*x)*D[y[x],x] + x*(1 + x)*D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to c_2 x^{1-b} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (a-2 b-\sqrt {a^2-2 a-4 c+1}+1\right ),\frac {1}{2} \left (a-2 b+\sqrt {a^2-2 a-4 c+1}+1\right ),2-b,-x\right )+c_1 \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (a-\sqrt {a^2-2 a-4 c+1}-1\right ),\frac {1}{2} \left (a+\sqrt {a^2-2 a-4 c+1}-1\right ),b,-x\right ) \]