76.13.44 problem 52

Internal problem ID [17555]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 4. Second order linear equations. Section 4.3 (Linear homogeneous equations with constant coefficients). Problems at page 239
Problem number : 52
Date solved : Monday, March 31, 2025 at 04:17:20 PM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} a \,x^{2} y^{\prime \prime }+b x y^{\prime }+c y&=0 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 69

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

\[ y = c_1 \,x^{\frac {-b +a +\sqrt {a^{2}+\left (-2 b -4 c \right ) a +b^{2}}}{2 a}}+c_2 \,x^{-\frac {b -a +\sqrt {a^{2}+\left (-2 b -4 c \right ) a +b^{2}}}{2 a}} \]

✓ Mathematica. Time used: 0.058 (sec). Leaf size: 102

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

\[ y(x)\to x^{-\frac {\sqrt {a} \sqrt {c} \sqrt {\frac {a^2-2 a (b+2 c)+b^2}{a c}}-a+b}{2 a}} \left (c_2 x^{\frac {\sqrt {c} \sqrt {\frac {a^2-2 a (b+2 c)+b^2}{a c}}}{\sqrt {a}}}+c_1\right ) \]

✓ Sympy. Time used: 1.131 (sec). Leaf size: 447

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

\[ \text {Solution too large to show} \]