61.28.17 problem 77

Internal problem ID [12577]
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 : 77
Date solved : Tuesday, January 28, 2025 at 03:22:04 AM
CAS classification : [[_2nd_order, _missing_y]]

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 115

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

\[ y = \frac {c_{2} a^{3}-\int \left (-c^{2} \left (\left (b^{3}+3 b^{2}+2 b \right ) \Gamma \left (b , -a x \right )-\Gamma \left (b +3\right )\right ) {\mathrm e}^{-a x} \left (-a x \right )^{-b}-{\mathrm e}^{-a x} x^{-b} c_{1} a^{3}+\left (\left (-b^{2}+\left (a x -3\right ) b -a^{2} x^{2}+2 a x -2\right ) c +a^{3} x \right ) c \right )d x}{a^{3}} \]

✓ Solution by Mathematica

Time used: 61.269 (sec). Leaf size: 98

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

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