Internal problem ID [10890]
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: 56.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]
\[ \boxed {y^{\prime \prime }+\left (x^{n} a +b \,x^{m}\right ) y^{\prime }+\left (a n \,x^{-1+n}+b m \,x^{m -1}\right ) y=0} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 65
dsolve(diff(y(x),x$2)+(a*x^n+b*x^m)*diff(y(x),x)+(a*n*x^(n-1)+b*m*x^(m-1))*y(x)=0,y(x), singsol=all)
\[ y = \left (c_{1} \left (\int {\mathrm e}^{\frac {a \,x^{n +1}}{n +1}+\frac {b \,x^{1+m}}{1+m}}d x \right )+c_{2} \right ) {\mathrm e}^{-\frac {a \,x^{n +1}}{n +1}-\frac {b \,x^{1+m}}{1+m}} \]
✓ Solution by Mathematica
Time used: 0.139 (sec). Leaf size: 74
DSolve[y''[x]+(a*x^n+b*x^m)*y'[x]+(a*n*x^(n-1)+b*m*x^(m-1))*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{x \left (-\frac {a x^n}{n+1}-\frac {b x^m}{m+1}\right )} \left (\int _1^x\exp \left (K[1] \left (\frac {b K[1]^m}{m+1}+\frac {a K[1]^n}{n+1}\right )\right ) c_1dK[1]+c_2\right ) \]