61.28.45 problem 105

Internal problem ID [12605]
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 : 105
Date solved : Tuesday, January 28, 2025 at 03:22:51 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

✓ Solution by Maple

Time used: 0.092 (sec). Leaf size: 78

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

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

✓ Solution by Mathematica

Time used: 0.362 (sec). Leaf size: 61

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

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