59.1.145 problem 147

Internal problem ID [9317]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 147
Date solved : Monday, January 27, 2025 at 06:01:24 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x^{2} \left (x^{2}+4\right ) y^{\prime \prime }+3 x \left (3 x^{2}+8\right ) y^{\prime }+\left (-9 x^{2}+1\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.114 (sec). Leaf size: 66

dsolve(4*x^2*(4+x^2)*diff(y(x),x$2)+3*x*(8+3*x^2)*diff(y(x),x)+(1-9*x^2)*y(x)=0,y(x), singsol=all)
 

\[ y = \frac {c_{2} \left (x^{2}+4\right )^{{5}/{8}} \left (x^{2} \operatorname {hypergeom}\left (\left [1, 1, \frac {13}{8}\right ], \left [2, 2\right ], -\frac {x^{2}}{4}\right )-\frac {32 \mcoloneq \gamma }{5}+\frac {64 \ln \left (2\right )}{5}-\frac {64 \ln \left (x \right )}{5}-\frac {32 \Psi \left (\frac {5}{8}\right )}{5}\right ) 2^{{3}/{4}}+c_{1} \left (x^{2}+4\right )^{{5}/{8}}-\frac {1024 c_{2}}{25}}{x^{{1}/{4}}} \]

✓ Solution by Mathematica

Time used: 0.273 (sec). Leaf size: 118

DSolve[4*x^2*(4+x^2)*D[y[x],{x,2}]+3*x*(8+3*x^2)*D[y[x],x]+(1-9*x^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \exp \left (\int _1^x\frac {17 K[1]^2+16}{8 K[1]^3+32 K[1]}dK[1]-\frac {1}{2} \int _1^x\frac {9 K[2]^2+24}{4 K[2]^3+16 K[2]}dK[2]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[3]}\frac {17 K[1]^2+16}{8 K[1]^3+32 K[1]}dK[1]\right )dK[3]+c_1\right ) \]