59.1.267 problem 270

Internal problem ID [9439]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 270
Date solved : Monday, January 27, 2025 at 06:02:50 PM
CAS classification : [_Jacobi]

\begin{align*} x \left (1-x \right ) y^{\prime \prime }+\left (\frac {3}{2}-2 x \right ) y^{\prime }-\frac {y}{4}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.023 (sec). Leaf size: 32

dsolve(x*(1-x)*diff(y(x),x$2)+(3/2-2*x)*diff(y(x),x)-1/4*y(x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.283 (sec). Leaf size: 104

DSolve[x*(1-x)*D[y[x],{x,2}]+(3/2-2*x)*D[y[x],x]-1/4*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

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