59.1.131 problem 133

Internal problem ID [9303]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 133
Date solved : Monday, January 27, 2025 at 06:01:14 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 36 x^{2} \left (1-2 x \right ) y^{\prime \prime }+24 x \left (1-9 x \right ) y^{\prime }+\left (1-70 x \right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.150 (sec). Leaf size: 93

dsolve(36*x^2*(1-2*x)*diff(y(x),x$2)+24*x*(1-9*x)*diff(y(x),x)+(1-70*x)*y(x)=0,y(x), singsol=all)
 

\[ y = \frac {x^{{1}/{6}} \left (2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (-1+2 x \right )^{{1}/{3}}}{-2+\left (-1+2 x \right )^{{1}/{3}}}\right ) c_{2} -2 \ln \left (1+\left (-1+2 x \right )^{{1}/{3}}\right ) c_{2} +\ln \left (1-\left (-1+2 x \right )^{{1}/{3}}+\left (-1+2 x \right )^{{2}/{3}}\right ) c_{2} +6 c_{2} \left (-1+2 x \right )^{{1}/{3}}+3 c_{1} \right )}{3 \left (-1+2 x \right )^{{4}/{3}}} \]

✓ Solution by Mathematica

Time used: 0.296 (sec). Leaf size: 112

DSolve[36*x^2*(1-2*x)*D[y[x],{x,2}]+24*x*(1-9*x)*D[y[x],x]+(1-70*x)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \exp \left (\int _1^x\frac {3-4 K[1]}{6 K[1]-12 K[1]^2}dK[1]-\frac {1}{2} \int _1^x\frac {2-18 K[2]}{3 K[2]-6 K[2]^2}dK[2]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[3]}\frac {3-4 K[1]}{6 K[1]-12 K[1]^2}dK[1]\right )dK[3]+c_1\right ) \]