Internal problem ID [10879]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-2 Equation of form
\(y''+f(x)y'+g(x)y=0\)
Problem number: 45.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }+a \,x^{n} y^{\prime }+b \,x^{-1+n} y=0} \]
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 119
dsolve(diff(y(x),x$2)+a*x^n*diff(y(x),x)+b*x^(n-1)*y(x)=0,y(x), singsol=all)
\[ y = c_{1} x \,{\mathrm e}^{-\frac {a \,x^{n +1}}{n +1}} \operatorname {KummerM}\left (\frac {a \left (n +1\right )-b}{a \left (n +1\right )}, \frac {2+n}{n +1}, \frac {a \,x^{n +1}}{n +1}\right )+c_{2} x \,{\mathrm e}^{-\frac {a \,x^{n +1}}{n +1}} \operatorname {KummerU}\left (\frac {a \left (n +1\right )-b}{a \left (n +1\right )}, \frac {2+n}{n +1}, \frac {a \,x^{n +1}}{n +1}\right ) \]
✓ Solution by Mathematica
Time used: 0.143 (sec). Leaf size: 120
DSolve[y''[x]+a*x^n*y'[x]+b*x^(n-1)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_2 \left (\frac {1}{n}+1\right )^{-\frac {1}{n+1}} n^{-\frac {1}{n+1}} a^{\frac {1}{n+1}} \left (x^n\right )^{\frac {1}{n}} \operatorname {Hypergeometric1F1}\left (\frac {a+b}{n a+a},\frac {n+2}{n+1},-\frac {a \left (x^n\right )^{1+\frac {1}{n}}}{n+1}\right )+c_1 \operatorname {Hypergeometric1F1}\left (\frac {b}{n a+a},\frac {n}{n+1},-\frac {a \left (x^n\right )^{1+\frac {1}{n}}}{n+1}\right ) \]