7.7.12 problem 12

Internal problem ID [2375]
Book : Differential equations and their applications, 3rd ed., M. Braun
Section : Section 2.2, linear equations with constant coefficients. Page 138
Problem number : 12
Date solved : Tuesday, September 30, 2025 at 05:35:02 AM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} t^{2} y^{\prime \prime }-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: 0.100 (sec). Leaf size: 25

ode:=t^2*diff(diff(y(t),t),t)-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);
 

\[ y = -\frac {\sqrt {3}\, t \left (-t^{\sqrt {3}}+t^{-\sqrt {3}}\right )}{6} \]

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 36

ode=t^2*D[y[t],{t,2}]-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^{1-\sqrt {3}} \left (t^{2 \sqrt {3}}-1\right )}{2 \sqrt {3}} \end{align*}

✓ Sympy. Time used: 0.117 (sec). Leaf size: 32

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(t**2*Derivative(y(t), (t, 2)) - 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)
 

\[ y{\left (t \right )} = - \frac {\sqrt {3}}{6 t^{-1 + \sqrt {3}}} + \frac {\sqrt {3} t^{1 + \sqrt {3}}}{6} \]