Internal problem ID [8719]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1141.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {a x y^{\prime \prime }+\left (x b +3 a \right ) y^{\prime }+3 y b=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 49
dsolve(a*x*diff(diff(y(x),x),x)+(b*x+3*a)*diff(y(x),x)+3*b*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} {\mathrm e}^{-\frac {b x}{a}}+c_{2} \left (\frac {\left (x b +a \right ) a}{x^{2}}+{\mathrm e}^{-\frac {b x}{a}} \operatorname {Ei}_{1}\left (-\frac {b x}{a}\right ) b^{2}\right ) \]
✓ Solution by Mathematica
Time used: 0.147 (sec). Leaf size: 57
DSolve[3*b*y[x] + (3*a + b*x)*y'[x] + a*x*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {e^{-\frac {b x}{a}} \left (2 a^2 c_1+b^2 c_2 \operatorname {ExpIntegralEi}\left (\frac {b x}{a}\right )\right )-\frac {a c_2 (a+b x)}{x^2}}{2 a^2} \\ \end{align*}