61.27.44 problem 54

Internal problem ID [12554]
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 : 54
Date solved : Tuesday, January 28, 2025 at 08:02:19 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+x^{n} \left (a \,x^{2}+\left (a c +b \right ) x +b c \right ) y^{\prime }-x^{n} \left (a x +b \right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 1.652 (sec). Leaf size: 78

dsolve(diff(y(x),x$2)+x^n*(a*x^2+(a*c+b)*x+b*c)*diff(y(x),x)-x^n*(a*x+b)*y(x)=0,y(x), singsol=all)
 

\[ y = -\left (x +c \right ) \left (\left (\int \frac {{\mathrm e}^{-\frac {x^{n +1} \left (a \,x^{2} \left (n +2\right ) \left (n +1\right )+\left (a c +b \right ) x \left (n +3\right ) \left (n +1\right )+b c \left (n +3\right ) \left (n +2\right )\right )}{\left (n +1\right ) \left (n +2\right ) \left (n +3\right )}}}{\left (x +c \right )^{2}}d x \right ) c_{1} +c_{2} \right ) \]

✗ Solution by Mathematica

Time used: 0.000 (sec). Leaf size: 0

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

Not solved