4.8.22 problem 36

Internal problem ID [1294]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Chapter 3, Second order linear equations, 3.3 Complex Roots of the Characteristic Equation , page 164
Problem number : 36
Date solved : Tuesday, September 30, 2025 at 04:32:16 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} t^{2} y^{\prime \prime }+4 t y^{\prime }+2 y&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 13

ode:=t^2*diff(diff(y(t),t),t)+4*t*diff(y(t),t)+2*y(t) = 0; 
dsolve(ode,y(t), singsol=all);
 

\[ y = \frac {c_2 t +c_1}{t^{2}} \]

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 34

ode=t^2*D[y[t],{t,2}]+4*t*D[y[t],t]+y[t]==0; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to t^{-\frac {3}{2}-\frac {\sqrt {5}}{2}} \left (c_2 t^{\sqrt {5}}+c_1\right ) \end{align*}

✓ Sympy. Time used: 0.101 (sec). Leaf size: 8

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(t**2*Derivative(y(t), (t, 2)) + 4*t*Derivative(y(t), t) + 2*y(t),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \frac {C_{1} + \frac {C_{2}}{t}}{t} \]