59.1.143 problem 145

Internal problem ID [9315]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 145
Date solved : Monday, January 27, 2025 at 06:01:22 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 x^{2} \left (x^{2}+2\right ) y^{\prime \prime }+7 x^{3} y^{\prime }+\left (3 x^{2}+1\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.219 (sec). Leaf size: 81

dsolve(2*x^2*(2+x^2)*diff(y(x),x$2)+7*x^3*diff(y(x),x)+(1+3*x^2)*y(x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.411 (sec). Leaf size: 93

DSolve[2*x^2*(2+x^2)*D[y[x],{x,2}]+7*x^3*D[y[x],x]+(1+3*x^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {\exp \left (\int _1^x\frac {3 K[1]^2+4}{4 K[1]^3+8 K[1]}dK[1]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[2]}\frac {3 K[1]^2+4}{4 K[1]^3+8 K[1]}dK[1]\right )dK[2]+c_1\right )}{\left (x^2+2\right )^{7/8}} \]