68.9.25 problem 36

Internal problem ID [17488]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.1, page 141
Problem number : 36
Date solved : Thursday, October 02, 2025 at 02:24:39 PM
CAS classification : [[_Emden, _Fowler]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 52

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

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

✓ Mathematica. Time used: 0.024 (sec). Leaf size: 57

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

\begin{align*} y(t)&\to t^{\frac {1}{2} \left (-\sqrt {a^2-2 a-4 b+1}-a+1\right )} \left (c_2 t^{\sqrt {a^2-2 a-4 b+1}}+c_1\right ) \end{align*}

✓ Sympy. Time used: 1.213 (sec). Leaf size: 617

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

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