61.27.28 problem 38

Internal problem ID [12538]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-2
Problem number : 38
Date solved : Tuesday, January 28, 2025 at 03:21:00 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 1.313 (sec). Leaf size: 294

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

\[ y = c_{1} {\mathrm e}^{-\frac {x \left (2 a^{2} b^{2} x^{2}+2 a b \,x^{2} \sqrt {b^{2} a^{2}}+3 a \,b^{2} x +3 b x \sqrt {b^{2} a^{2}}+12 a \sqrt {b^{2} a^{2}}\right )}{12 \sqrt {b^{2} a^{2}}}} \operatorname {HeunT}\left (\frac {b 3^{{2}/{3}}}{2 \left (b^{2} a^{2}\right )^{{1}/{3}}}, -\frac {6 a b}{\sqrt {b^{2} a^{2}}}, -\frac {b^{2} 3^{{1}/{3}}}{4 \left (b^{2} a^{2}\right )^{{2}/{3}}}, \frac {3^{{2}/{3}} a \,b^{2} \left (2 a x +1\right )}{6 \left (b^{2} a^{2}\right )^{{5}/{6}}}\right )+c_{2} {\mathrm e}^{-\frac {x \left (-2 a^{2} b^{2} x^{2}+2 a b \,x^{2} \sqrt {b^{2} a^{2}}-3 a \,b^{2} x +3 b x \sqrt {b^{2} a^{2}}+12 a \sqrt {b^{2} a^{2}}\right )}{12 \sqrt {b^{2} a^{2}}}} \operatorname {HeunT}\left (\frac {b 3^{{2}/{3}}}{2 \left (b^{2} a^{2}\right )^{{1}/{3}}}, \frac {6 a b}{\sqrt {b^{2} a^{2}}}, -\frac {b^{2} 3^{{1}/{3}}}{4 \left (b^{2} a^{2}\right )^{{2}/{3}}}, -\frac {3^{{2}/{3}} a \left (a x +\frac {1}{2}\right ) b^{2}}{3 \left (b^{2} a^{2}\right )^{{5}/{6}}}\right ) \]

✓ Solution by Mathematica

Time used: 0.766 (sec). Leaf size: 82

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

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