59.1.98 problem 100

Internal problem ID [9270]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 100
Date solved : Monday, January 27, 2025 at 06:00:48 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (x +2\right ) y^{\prime \prime }+5 x \left (1-x \right ) y^{\prime }-\left (2-8 x \right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.113 (sec). Leaf size: 113

dsolve(x^2*(2+x)*diff(y(x),x$2)+5*x*(1-x)*diff(y(x),x)-(2-8*x)*y(x)=0,y(x), singsol=all)
 

\[ y = \frac {c_{1} \left (40 x^{4}-160 x^{3}+60 x^{2}+8 x +1\right )}{x^{2}}+\frac {4 \left (1050 \left (x^{4}-4 x^{3}+\frac {3}{2} x^{2}+\frac {1}{5} x +\frac {1}{40}\right ) x^{{3}/{2}} \operatorname {arcsinh}\left (\frac {\sqrt {x}\, \sqrt {2}}{2}\right )+\sqrt {x +2}\, x^{2} \left (x^{5}+41 x^{4}-\frac {6987}{4} x^{3}+\frac {13367}{4} x^{2}-\frac {805}{8} x -\frac {105}{8}\right )\right ) \left (-x -2\right )^{{3}/{4}} c_{2}}{105 \left (x +2\right )^{{3}/{4}} x^{{7}/{2}}} \]

✓ Solution by Mathematica

Time used: 0.773 (sec). Leaf size: 163

DSolve[x^2*(2+x)*D[y[x],{x,2}]+5*x*(1-x)*D[y[x],x]-(2-8*x)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {1}{40} \left (40 x^4-160 x^3+60 x^2+8 x+1\right ) \exp \left (\int _1^x-\frac {9 K[1]+3}{2 K[1]^2+4 K[1]}dK[1]-\frac {1}{2} \int _1^x\frac {5-5 K[2]}{K[2]^2+2 K[2]}dK[2]\right ) \left (c_2 \int _1^x\frac {1600 \exp \left (-2 \int _1^{K[3]}-\frac {9 K[1]+3}{2 K[1]^2+4 K[1]}dK[1]\right )}{\left (40 K[3]^4-160 K[3]^3+60 K[3]^2+8 K[3]+1\right )^2}dK[3]+c_1\right ) \]