59.1.345 problem 352

Internal problem ID [9517]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 352
Date solved : Monday, January 27, 2025 at 06:03:40 PM
CAS classification : [_Jacobi]

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

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 53

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

\[ y = \left (-12 x -3\right ) c_{2} \ln \left (2 x -1+2 \sqrt {\left (x -1\right ) x}\right )+\left (4 x +26\right ) c_{2} \sqrt {\left (x -1\right ) x}+4 \left (x +\frac {1}{4}\right ) \left (3 c_{2} \ln \left (2\right )+c_{1} \right ) \]

✓ Solution by Mathematica

Time used: 0.532 (sec). Leaf size: 130

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

\[ y(x)\to \frac {1}{4} (4 x+1) \exp \left (\int _1^x\frac {4 K[1]+1}{4 K[1]-4 K[1]^2}dK[1]-\frac {1}{2} \int _1^x\frac {4 K[2]+1}{2 K[2]-2 K[2]^2}dK[2]\right ) \left (c_2 \int _1^x\frac {16 \exp \left (-2 \int _1^{K[3]}\frac {4 K[1]+1}{4 K[1]-4 K[1]^2}dK[1]\right )}{(4 K[3]+1)^2}dK[3]+c_1\right ) \]