59.1.138 problem 140

Internal problem ID [9310]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 140
Date solved : Monday, January 27, 2025 at 06:01:19 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (1-x \right ) y^{\prime \prime }+x \left (7+x \right ) y^{\prime }+\left (9-x \right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 72

dsolve(x^2*(1-x)*diff(y(x),x$2)+x*(7+x)*diff(y(x),x)+(9-x)*y(x)=0,y(x), singsol=all)
 

\[ y = \frac {3 c_{2} \left (x^{4}+16 x^{3}+36 x^{2}+16 x +1\right ) \ln \left (x \right )+c_{1} x^{4}+\left (16 c_{1} +120 c_{2} \right ) x^{3}+\left (36 c_{1} +450 c_{2} \right ) x^{2}+\left (16 c_{1} +280 c_{2} \right ) x +c_{1} +25 c_{2}}{x^{3}} \]

✓ Solution by Mathematica

Time used: 0.751 (sec). Leaf size: 145

DSolve[x^2*(1-x)*D[y[x],{x,2}]+x*(7+x)*D[y[x],x]+(9-x)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \left (x^4+16 x^3+36 x^2+16 x+1\right ) \exp \left (\int _1^x\left (\frac {1}{2 K[1]}-\frac {4}{K[1]-1}\right )dK[1]-\frac {1}{2} \int _1^x\frac {K[2]+7}{K[2]-K[2]^2}dK[2]\right ) \left (c_2 \int _1^x\frac {\exp \left (-2 \int _1^{K[3]}\left (\frac {1}{2 K[1]}-\frac {4}{K[1]-1}\right )dK[1]\right )}{\left (K[3]^4+16 K[3]^3+36 K[3]^2+16 K[3]+1\right )^2}dK[3]+c_1\right ) \]