61.27.46 problem 56

Internal problem ID [12556]
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 : 56
Date solved : Tuesday, January 28, 2025 at 03:21:29 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

✓ Solution by Maple

Time used: 0.231 (sec). Leaf size: 72

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)
 

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

✓ Solution by Mathematica

Time used: 60.071 (sec). Leaf size: 74

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

\[ y(x)\to e^{x \left (-\frac {a x^n}{n+1}-\frac {b x^m}{m+1}\right )} \left (\int _1^x\exp \left (K[1] \left (\frac {b K[1]^m}{m+1}+\frac {a K[1]^n}{n+1}\right )\right ) c_1dK[1]+c_2\right ) \]