8.7.11 problem 11

Internal problem ID [2555]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.2. Linear equations with constant coefficients. Excercises page 140
Problem number : 11
Date solved : Tuesday, September 30, 2025 at 05:47:06 AM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} t^{2} y^{\prime \prime }+5 t y^{\prime }-2 y&=0 \end{align*}

With initial conditions

\begin{align*} y \left (1\right )&=0 \\ y^{\prime }\left (1\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.097 (sec). Leaf size: 27

ode:=t^2*diff(diff(y(t),t),t)+5*t*diff(y(t),t)-2*y(t) = 0; 
ic:=[y(1) = 0, D(y)(1) = 1]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

\[ y = \frac {\sqrt {6}\, \left (t^{\sqrt {6}}-t^{-\sqrt {6}}\right )}{12 t^{2}} \]

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 36

ode=t^2*D[y[t],{t,2}]+5*t*D[y[t],t]-2*y[t]==0; 
ic={y[1]==0,Derivative[1][y][1] ==1}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {t^{-2-\sqrt {6}} \left (t^{2 \sqrt {6}}-1\right )}{2 \sqrt {6}} \end{align*}

✓ Sympy. Time used: 0.129 (sec). Leaf size: 32

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(t**2*Derivative(y(t), (t, 2)) + 5*t*Derivative(y(t), t) - 2*y(t),0) 
ics = {y(1): 0, Subs(Derivative(y(t), t), t, 1): 1} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \frac {\sqrt {6} t^{-2 + \sqrt {6}}}{12} - \frac {\sqrt {6}}{12 t^{2 + \sqrt {6}}} \]