60.3.255 problem 1260

Internal problem ID [11265]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1260
Date solved : Monday, January 27, 2025 at 11:00:57 PM
CAS classification : [[_2nd_order, _missing_y]]

\begin{align*} x \left (x -1\right ) y^{\prime \prime }+\left (\left (\operatorname {a1} +\operatorname {b1} +1\right ) x -\operatorname {d1} \right ) y^{\prime }+\operatorname {a1} \operatorname {b1} \operatorname {d1}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 81

dsolve(x*(x-1)*diff(diff(y(x),x),x)+((a1+b1+1)*x-d1)*diff(y(x),x)+a1*b1*d1=0,y(x), singsol=all)
 

\[ y = -\int \left (x -1\right )^{-\operatorname {a1} -\operatorname {b1} -1+\operatorname {d1}} x^{-\operatorname {d1}} \left (\operatorname {b1} \operatorname {signum}\left (x -1\right )^{\operatorname {a1} +\operatorname {b1} -\operatorname {d1}} \left (-\operatorname {signum}\left (x -1\right )\right )^{-\operatorname {a1} -\operatorname {b1} +\operatorname {d1}} \operatorname {a1} \,x^{\operatorname {d1}} \operatorname {hypergeom}\left (\left [\operatorname {d1} , -\operatorname {a1} -\operatorname {b1} +\operatorname {d1} \right ], \left [1+\operatorname {d1} \right ], x\right )-c_{1} \right )d x +c_{2} \]

✓ Solution by Mathematica

Time used: 2.752 (sec). Leaf size: 106

DSolve[a1*b1*d1 + (-d1 + (1 + a1 + b1)*x)*D[y[x],x] + (-1 + x)*x*D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \int _1^x\exp \left (\int _1^{K[3]}\frac {\text {d1}-(\text {a1}+\text {b1}+1) K[1]}{(K[1]-1) K[1]}dK[1]\right ) \left (c_1+\int _1^{K[3]}-\frac {\text {a1} \text {b1} \text {d1} \exp \left (-\int _1^{K[2]}\frac {\text {d1}-(\text {a1}+\text {b1}+1) K[1]}{(K[1]-1) K[1]}dK[1]\right )}{(K[2]-1) K[2]}dK[2]\right )dK[3]+c_2 \]