61.27.39 problem 49

Internal problem ID [12549]
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 : 49
Date solved : Tuesday, January 28, 2025 at 08:02:16 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+2 a \,x^{n} y^{\prime }+\left (a^{2} x^{2 n}+b \,x^{2 m}+a n \,x^{n -1}+c \,x^{m -1}\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 1.621 (sec). Leaf size: 142

dsolve(diff(y(x),x$2)+2*a*x^n*diff(y(x),x)+(a^2*x^(2*n)+b*x^(2*m)+a*n*x^(n-1)+c*x^(m-1))*y(x)=0,y(x), singsol=all)
 

\[ y = x \left (\operatorname {KummerU}\left (\frac {\left (m +2\right ) \sqrt {b}+i c}{\sqrt {b}\, \left (2+2 m \right )}, \frac {m +2}{m +1}, \frac {2 i \sqrt {b}\, x^{m +1}}{m +1}\right ) c_{2} +\operatorname {KummerM}\left (\frac {\left (m +2\right ) \sqrt {b}+i c}{\sqrt {b}\, \left (2+2 m \right )}, \frac {m +2}{m +1}, \frac {2 i \sqrt {b}\, x^{m +1}}{m +1}\right ) c_{1} \right ) {\mathrm e}^{\frac {-i \sqrt {b}\, \left (n +1\right ) x^{m +1}-a \,x^{n +1} \left (m +1\right )}{\left (n +1\right ) \left (m +1\right )}} \]

✓ Solution by Mathematica

Time used: 0.215 (sec). Leaf size: 236

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

\[ y(x)\to 2^{\frac {m}{2 m+2}} x^{-m/2} \left (x^{m+1}\right )^{\frac {m}{2 m+2}} \exp \left (-x \left (\frac {a x^n}{n+1}+\frac {\sqrt {b} x^m}{\sqrt {-(m+1)^2}}\right )\right ) \left (c_1 \operatorname {HypergeometricU}\left (-\frac {(m+1) \left (m c+c+\sqrt {b} m \sqrt {-(m+1)^2}\right )}{2 \sqrt {b} \left (-(m+1)^2\right )^{3/2}},\frac {m}{m+1},\frac {2 \sqrt {b} x^{m+1}}{\sqrt {-(m+1)^2}}\right )+c_2 L_{\frac {(m+1) \left (m c+c+\sqrt {b} m \sqrt {-(m+1)^2}\right )}{2 \sqrt {b} \left (-(m+1)^2\right )^{3/2}}}^{-\frac {1}{m+1}}\left (\frac {2 \sqrt {b} x^{m+1}}{\sqrt {-(m+1)^2}}\right )\right ) \]