61.28.19 problem 79

Internal problem ID [12579]
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 : 79
Date solved : Tuesday, January 28, 2025 at 08:02:31 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x y^{\prime \prime }+\left (a b \,x^{2}+b -5\right ) y^{\prime }+2 a^{2} \left (b -2\right ) x^{3} y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.506 (sec). Leaf size: 92

dsolve(x*diff(y(x),x$2)+(a*b*x^2+b-5)*diff(y(x),x)+2*a^2*(b-2)*x^3*y(x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.658 (sec). Leaf size: 94

DSolve[x*D[y[x],{x,2}]+(a*b*x^2+b-5)*D[y[x],x]+2*a^2*(b-2)*x^3*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to e^{-2 a^2 \int \frac {x^3}{a x^2+1} \, dx} \left (c_2 \int _1^x\exp \left (\int _1^{K[2]}\frac {4 a^2 K[1]^4+5 a K[1]^2-b \left (a K[1]^2+1\right )^2+5}{a K[1]^3+K[1]}dK[1]\right )dK[2]+c_1\right ) \]