Internal problem ID [11037]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-7 Equation of form
\((a_4 x^4+a_3 x^3+a_2 x^2 x+a_1 x+a_0) y''+f(x)y'+g(x)y=0\)
Problem number: 213.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{4} y^{\prime \prime }-\left (a +b \right ) x^{2} y^{\prime }+\left (\left (a +b \right ) x +a b \right ) y=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 25
dsolve(x^4*diff(y(x),x$2)-(a+b)*x^2*diff(y(x),x)+((a+b)*x+a*b)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = x \left ({\mathrm e}^{-\frac {a}{x}} c_{1} +{\mathrm e}^{-\frac {b}{x}} c_{2} \right ) \]
✓ Solution by Mathematica
Time used: 0.214 (sec). Leaf size: 37
DSolve[x^4*y''[x]-(a+b)*x^2*y'[x]+((a+b)*x+a*b)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {c_2 x e^{-\frac {a}{x}}}{a-b}+c_1 x e^{-\frac {b}{x}} \]