Internal problem ID [10889]
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: 55.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }+\left (x^{n} a +b \,x^{m}\right ) y^{\prime }-\left (a \,x^{-1+n}+x^{m -1} b \right ) y=0} \]
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 54
dsolve(diff(y(x),x$2)+(a*x^n+b*x^m)*diff(y(x),x)-(a*x^(n-1)+b*x^(m-1))*y(x)=0,y(x), singsol=all)
\[ y = c_{1} x +c_{2} x \left (\int \frac {{\mathrm e}^{-\frac {x \left (x^{n} a m +x^{m} b n +a \,x^{n}+x^{m} b \right )}{\left (n +1\right ) \left (1+m \right )}}}{x^{2}}d x \right ) \]
✓ Solution by Mathematica
Time used: 1.216 (sec). Leaf size: 55
DSolve[y''[x]+(a*x^n+b*x^m)*y'[x]-(a*x^(n-1)+b*x^(m-1))*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to x \left (c_2 \int _1^x\frac {\exp \left (K[1] \left (-\frac {b K[1]^m}{m+1}-\frac {a K[1]^n}{n+1}\right )\right )}{K[1]^2}dK[1]+c_1\right ) \]