13.7.10 problem 10

Internal problem ID [2373]
Book : Differential equations and their applications, 3rd ed., M. Braun
Section : Section 2.2, linear equations with constant coefficients. Page 138
Problem number : 10
Date solved : Tuesday, March 04, 2025 at 02:08:33 PM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} t^{2} y^{\prime \prime }+\alpha t y^{\prime }+\beta y&=0 \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 52

ode:=t^2*diff(diff(y(t),t),t)+alpha*t*diff(y(t),t)+beta*y(t) = 0; 
dsolve(ode,y(t), singsol=all);
 

\[ y = t^{-\frac {\alpha }{2}} \sqrt {t}\, \left (t^{\frac {\sqrt {\alpha ^{2}-2 \alpha -4 \beta +1}}{2}} c_1 +t^{-\frac {\sqrt {\alpha ^{2}-2 \alpha -4 \beta +1}}{2}} c_2 \right ) \]

✓ Mathematica. Time used: 0.049 (sec). Leaf size: 57

ode=t^2*D[y[t],{t,2}]+\[Alpha]*t*D[y[t],t]+\[Beta]*y[t]==0; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to t^{\frac {1}{2} \left (-\sqrt {\alpha ^2-2 \alpha -4 \beta +1}-\alpha +1\right )} \left (c_2 t^{\sqrt {\alpha ^2-2 \alpha -4 \beta +1}}+c_1\right ) \]

✓ Sympy. Time used: 2.241 (sec). Leaf size: 617

from sympy import * 
t = symbols("t") 
Alpha = symbols("Alpha") 
BETA = symbols("BETA") 
y = Function("y") 
ode = Eq(Alpha*t*Derivative(y(t), t) + BETA*y(t) + t**2*Derivative(y(t), (t, 2)),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

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