61.28.22 problem 82

Internal problem ID [12582]
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 : 82
Date solved : Tuesday, January 28, 2025 at 03:22:13 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

✓ Solution by Maple

Time used: 0.430 (sec). Leaf size: 46

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

\[ y = x^{-c +1} {\mathrm e}^{-\frac {x \left (a x +2 b \right )}{2}} \left (c_{1} \left (\int x^{c -2} {\mathrm e}^{\frac {1}{2} a \,x^{2}+b x}d x \right )+c_{2} \right ) \]

✓ Solution by Mathematica

Time used: 0.393 (sec). Leaf size: 68

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

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