60.3.285 problem 1302

Internal problem ID [11281]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1302
Date solved : Sunday, March 30, 2025 at 08:07:49 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \operatorname {A2} \left (a x +b \right )^{2} y^{\prime \prime }+\operatorname {A1} \left (a x +b \right ) y^{\prime }+\operatorname {A0} \left (a x +b \right ) y&=0 \end{align*}

✓ Maple. Time used: 0.027 (sec). Leaf size: 98

ode:=A2*(a*x+b)^2*diff(diff(y(x),x),x)+A1*(a*x+b)*diff(y(x),x)+A0*(a*x+b)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (a x +b \right )^{-\frac {-a \operatorname {A2} +\operatorname {A1}}{2 a \operatorname {A2}}} \left (\operatorname {BesselY}\left (\frac {a \operatorname {A2} -\operatorname {A1}}{a \operatorname {A2}}, 2 \sqrt {\operatorname {A0}}\, \sqrt {\frac {a x +b}{a^{2} \operatorname {A2}}}\right ) c_2 +\operatorname {BesselJ}\left (\frac {a \operatorname {A2} -\operatorname {A1}}{a \operatorname {A2}}, 2 \sqrt {\operatorname {A0}}\, \sqrt {\frac {a x +b}{a^{2} \operatorname {A2}}}\right ) c_1 \right ) \]

✓ Mathematica. Time used: 0.152 (sec). Leaf size: 165

ode=A0*(b + a*x)*y[x] + A1*(b + a*x)*D[y[x],x] + A2*(b + a*x)^2*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to (-1)^{-\frac {\text {A1}}{a \text {A2}}} \left (\frac {b}{a}+x\right )^{\frac {\text {A1}}{2 a \text {A2}}} (\text {A2} (a x+b))^{-\frac {\text {A1}}{2 a \text {A2}}} \left (-\frac {\text {A0} (a x+b)}{a^2 \text {A2}}\right )^{\frac {1}{2}-\frac {\text {A1}}{2 a \text {A2}}} \left (c_1 (-1)^{\frac {\text {A1}}{a \text {A2}}} \operatorname {BesselI}\left (\frac {\text {A1}}{a \text {A2}}-1,2 \sqrt {-\frac {\text {A0} (b+a x)}{a^2 \text {A2}}}\right )-c_2 K_{\frac {\text {A1}}{a \text {A2}}-1}\left (2 \sqrt {-\frac {\text {A0} (b+a x)}{a^2 \text {A2}}}\right )\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
A0 = symbols("A0") 
A1 = symbols("A1") 
A2 = symbols("A2") 
a = symbols("a") 
b = symbols("b") 
y = Function("y") 
ode = Eq(A0*(a*x + b)*y(x) + A1*(a*x + b)*Derivative(y(x), x) + A2*(a*x + b)**2*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False