61.28.28 problem 88

Internal problem ID [12588]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-3
Problem number : 88
Date solved : Tuesday, January 28, 2025 at 03:22:24 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 1.053 (sec). Leaf size: 143

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

\[ y = \frac {c_{1} {\mathrm e}^{-\frac {x^{2} \operatorname {csgn}\left (a \right ) \left (\operatorname {csgn}\left (a \right )+1\right ) \left (a x +\frac {3 b}{2}\right )}{6}} \operatorname {HeunT}\left (\frac {3^{{2}/{3}} b}{2 \left (a^{2}\right )^{{1}/{3}}}, -6 \,\operatorname {csgn}\left (a \right ), -\frac {b^{2} 3^{{1}/{3}}}{4 \left (a^{2}\right )^{{2}/{3}}}, \frac {3^{{2}/{3}} a \left (2 a x +b \right )}{6 \left (a^{2}\right )^{{5}/{6}}}\right )+c_{2} {\mathrm e}^{-\frac {x^{2} \operatorname {csgn}\left (a \right ) \left (\operatorname {csgn}\left (a \right )-1\right ) \left (a x +\frac {3 b}{2}\right )}{6}} \operatorname {HeunT}\left (\frac {3^{{2}/{3}} b}{2 \left (a^{2}\right )^{{1}/{3}}}, 6 \,\operatorname {csgn}\left (a \right ), -\frac {b^{2} 3^{{1}/{3}}}{4 \left (a^{2}\right )^{{2}/{3}}}, -\frac {3^{{2}/{3}} a \left (2 a x +b \right )}{6 \left (a^{2}\right )^{{5}/{6}}}\right )}{x} \]

✓ Solution by Mathematica

Time used: 0.439 (sec). Leaf size: 76

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

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