Internal problem ID [10929]
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 Equation of form
\((a x + b)y''+f(x)y'+g(x)y=0\)
Problem number: 105.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _exact, _linear, _homogeneous]]
\[ \boxed {\left (x +a \right ) y^{\prime \prime }+\left (b x +c \right ) y^{\prime }+y b=0} \]
✓ Solution by Maple
Time used: 0.031 (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)
\[ y \left (x \right ) = -\left (-\left (a +x \right )^{a b -c +1} c_{1} +\left (\Gamma \left (-a b +c \right )+\Gamma \left (-a b +c -1, -b \left (a +x \right )\right ) \left (a b -c +1\right )\right ) b \left (a +x \right ) c_{2} \left (-b \left (a +x \right )\right )^{a b -c}\right ) {\mathrm e}^{-b x} \]
✓ Solution by Mathematica
Time used: 0.559 (sec). Leaf size: 90
DSolve[(x+a)*y''[x]+(b*x+c)*y'[x]+b*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-b (a+x)} (a+x)^{1-c} (-b (a+x))^{-c} \left (c_1 e^{a b} (a+x)^{a b} (-b (a+x))^c+b c_2 (-b (a+x))^{a b} (a+x)^c \Gamma (-a b+c-1,-b (a+x))\right ) \]