61.27.45 problem 55

Internal problem ID [12555]
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
Problem number : 55
Date solved : Tuesday, January 28, 2025 at 03:21:27 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+\left (a \,x^{n}+b \,x^{m}\right ) y^{\prime }-\left (a \,x^{n -1}+b \,x^{m -1}\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 1.172 (sec). Leaf size: 47

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)
 

\[ y = x \left (c_{1} +c_{2} \left (\int \frac {{\mathrm e}^{-\frac {\left (b \left (n +1\right ) x^{m}+a \,x^{n} \left (m +1\right )\right ) x}{\left (n +1\right ) \left (m +1\right )}}}{x^{2}}d x \right )\right ) \]

✓ Solution by Mathematica

Time used: 0.818 (sec). Leaf size: 55

DSolve[D[y[x],{x,2}]+(a*x^n+b*x^m)*D[y[x],x]-(a*x^(n-1)+b*x^(m-1))*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to x \left (c_2 \int _1^x\frac {\exp \left (K[1] \left (-\frac {b K[1]^m}{m+1}-\frac {a K[1]^n}{n+1}\right )\right )}{K[1]^2}dK[1]+c_1\right ) \]