61.27.31 problem 41

Internal problem ID [12462]
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 : Wednesday, March 05, 2025 at 07:07:01 PM
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*}

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

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

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

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

ode=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; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.947 (sec). Leaf size: 3

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
y = Function("y") 
ode = Eq((a*x**3 + 2*b)*Derivative(y(x), x) + (a*b*x**3 - a*x**2 + b**2)*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = O\left (1\right ) \]