60.3.280 problem 1285

Internal problem ID [11290]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1285
Date solved : Tuesday, January 28, 2025 at 05:57:57 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x \left (4 x -1\right ) y^{\prime \prime }+\left (\left (4 a +2\right ) x -a \right ) y^{\prime }+a \left (a -1\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.024 (sec). Leaf size: 56

dsolve(x*(4*x-1)*diff(diff(y(x),x),x)+((4*a+2)*x-a)*diff(y(x),x)+a*(a-1)*y(x)=0,y(x), singsol=all)
 

\[ y = c_{1} 2^{a -1} \left (1+\sqrt {1-4 x}\right )^{-a +1}+c_{2} x^{-a +1} 2^{-a +1} \left (1+\sqrt {1-4 x}\right )^{a -1} \]

✓ Solution by Mathematica

Time used: 16.049 (sec). Leaf size: 210

DSolve[(-1 + a)*a*y[x] + (-a + (2 + 4*a)*x)*D[y[x],x] + x*(-1 + 4*x)*D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
 

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