Internal problem ID [10891]
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: 57.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }+\left (x^{n} a +b \,x^{m}\right ) y^{\prime }+\left (a \left (n +1\right ) x^{-1+n}+b \left (m +1\right ) x^{m -1}\right ) y=0} \]
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 148
dsolve(diff(y(x),x$2)+(a*x^n+b*x^m)*diff(y(x),x)+(a*(n+1)*x^(n-1)+b*(m+1)*x^(m-1))*y(x)=0,y(x), singsol=all)
\[ y = c_{1} {\mathrm e}^{-\frac {a m \,x^{n +1}+b n \,x^{1+m}+a \,x^{n +1}+b \,x^{1+m}}{\left (n +1\right ) \left (1+m \right )}} x +c_{2} {\mathrm e}^{-\frac {a m \,x^{n +1}+b n \,x^{1+m}+a \,x^{n +1}+b \,x^{1+m}}{\left (n +1\right ) \left (1+m \right )}} \left (\int \frac {{\mathrm e}^{\frac {a m \,x^{n +1}+b n \,x^{1+m}+a \,x^{n +1}+b \,x^{1+m}}{\left (n +1\right ) \left (1+m \right )}}}{x^{2}}d x \right ) x \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y''[x]+(a*x^n+b*x^m)*y'[x]+(a*(n+1)*x^(n-1)+b*(m+1)*x^(m-1))*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
Not solved