60.3.250 problem 1255

Internal problem ID [11260]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1255
Date solved : Monday, January 27, 2025 at 11:00:49 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} x \left (x -1\right ) y^{\prime \prime }+a y^{\prime }-2 y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 42

dsolve(x*(x-1)*diff(diff(y(x),x),x)+a*diff(y(x),x)-2*y(x)=0,y(x), singsol=all)
 

\[ y = x^{a +1} c_{2} \left (x -1\right )^{-a +1}+c_{1} \left (a^{2}+a \left (-1+2 x \right )+2 x^{2}-2 x \right ) \]

✓ Solution by Mathematica

Time used: 0.566 (sec). Leaf size: 129

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

\[ y(x)\to \frac {\left (a^2+a (2 x-1)+2 (x-1) x\right ) \left (c_2 \int _1^x\exp \left (\int _1^{K[2]}-\frac {a^3+(2 K[1]-1) a^2+6 (K[1]-1) K[1] a+4 K[1] \left (2 K[1]^2-3 K[1]+1\right )}{(K[1]-1) K[1] \left (a^2+(2 K[1]-1) a+2 (K[1]-1) K[1]\right )}dK[1]\right )dK[2]+c_1\right )}{a^2+3 a+4} \]