61.30.2 problem 150

Internal problem ID [12571]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-5
Problem number : 150
Date solved : Monday, March 31, 2025 at 05:39:35 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (-a^{2}+x^{2}\right ) y^{\prime \prime }+b y^{\prime }-6 y&=0 \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 76

ode:=(-a^2+x^2)*diff(diff(y(x),x),x)+b*diff(y(x),x)-6*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\frac {c_1 \left (10 a^{2} b +24 a^{2} x -b^{3}-6 b^{2} x -18 b \,x^{2}-24 x^{3}\right )}{24}+c_2 \left (a -x \right ) \left (a +x \right ) \left (b -4 x \right ) \left (\frac {a +x}{a -x}\right )^{\frac {b}{2 a}} \]

✓ Mathematica. Time used: 2.092 (sec). Leaf size: 385

ode=(x^2-a^2)*D[y[x],{x,2}]+b*D[y[x],x]-6*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \exp \left (\int _1^x\frac {-16 (b+12 K[1]) a^4-2 (b+4 K[1]) \left (b^2-96 K[1]^2\right ) a^2+2 K[1] \left (b^4+3 K[1] b^3-12 K[1]^2 b^2-120 K[1]^3 b-288 K[1]^4\right )+\sqrt {\left (b^5-20 a^2 b^3+64 a^4 b\right )^2}}{2 (b-4 K[1]) (K[1]-a) (a+K[1]) \left (b^3+6 K[1] b^2+18 K[1]^2 b+24 K[1]^3-2 a^2 (5 b+12 K[1])\right )}dK[1]-\frac {1}{2} \int _1^x-\frac {b}{a^2-K[2]^2}dK[2]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[3]}\frac {-16 (b+12 K[1]) a^4-2 (b+4 K[1]) \left (b^2-96 K[1]^2\right ) a^2+2 K[1] \left (b^4+3 K[1] b^3-12 K[1]^2 b^2-120 K[1]^3 b-288 K[1]^4\right )+\sqrt {\left (b^5-20 a^2 b^3+64 a^4 b\right )^2}}{2 (b-4 K[1]) (K[1]-a) (a+K[1]) \left (b^3+6 K[1] b^2+18 K[1]^2 b+24 K[1]^3-2 a^2 (5 b+12 K[1])\right )}dK[1]\right )dK[3]+c_1\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
y = Function("y") 
ode = Eq(b*Derivative(y(x), x) + (-a**2 + x**2)*Derivative(y(x), (x, 2)) - 6*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False