Internal problem ID [11018]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-6 Equation of form
\((a_3 x^3+a_2 x^2 x+a_1 x+a_0) y''+f(x)y'+g(x)y=0\)
Problem number: 184.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{3} y^{\prime \prime }+\left (a \,x^{2}+b x \right ) y^{\prime }+b y=0} \]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 118
dsolve(x^3*diff(y(x),x$2)+(a*x^2+b*x)*diff(y(x),x)+b*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {c_{1} \left (-\left (a -2\right ) x \left (-1\right )^{-a} \left (\Gamma \left (a , -\frac {b}{x}\right )-\Gamma \left (a \right )\right ) b^{-\frac {a}{2}+1}-\left (-1\right )^{-a} \left (\Gamma \left (a , -\frac {b}{x}\right )-\Gamma \left (a \right )\right ) b^{-\frac {a}{2}+2}-{\mathrm e}^{\frac {b}{x}} x \left (b^{\frac {a}{2}+1} x^{-a}-x^{1-a} b^{\frac {a}{2}}\right )\right )}{x}+\frac {c_{2} \left (x \left (a -2\right )+b \right )}{x} \]
✓ Solution by Mathematica
Time used: 2.653 (sec). Leaf size: 62
DSolve[x^3*y''[x]+(a*x^2+b*x)*y'[x]+b*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {((a-2) x+b) \left (c_2 \int _1^x\frac {e^{\frac {b}{K[1]}} K[1]^{2-a}}{(b+(a-2) K[1])^2}dK[1]+c_1\right )}{x (a+b-2)} \]