61.27.31 problem 41

Internal problem ID [12541]
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 : 41
Date solved : Tuesday, January 28, 2025 at 03:21:06 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+\left (a \,x^{3}+2 b \right ) y^{\prime }+\left (a b \,x^{3}-a \,x^{2}+b^{2}\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.069 (sec). Leaf size: 85

dsolve(diff(y(x),x$2)+(a*x^3+2*b)*diff(y(x),x)+(a*b*x^3-a*x^2+b^2)*y(x)=0,y(x), singsol=all)
 

\[ y = \frac {\frac {7 \,2^{{1}/{4}} c_{2} a \left (a \,x^{4}\right )^{{3}/{8}} \left (a \,x^{4}+3\right ) {\mathrm e}^{-\frac {x \left (a \,x^{3}+4 b \right )}{4}}}{8}+\operatorname {WhittakerM}\left (\frac {3}{8}, \frac {7}{8}, \frac {a \,x^{4}}{4}\right ) {\mathrm e}^{-\frac {x \left (a \,x^{3}+8 b \right )}{8}} c_{2} a^{2} x^{4}+{\mathrm e}^{-b x} c_{1} x^{{5}/{2}}}{x^{{3}/{2}}} \]

✓ Solution by Mathematica

Time used: 0.408 (sec). Leaf size: 56

DSolve[D[y[x],{x,2}]+(a*x^3+2*b)*D[y[x],x]+(a*b*x^3-a*x^2+b^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {1}{8} e^{-b x-2} \left (8 e^2 c_1 x-\sqrt {2} c_2 \sqrt [4]{a x^4} \Gamma \left (-\frac {1}{4},\frac {a x^4}{4}\right )\right ) \]