14.10.10 problem Problem 23

Internal problem ID [3782]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 8, Linear differential equations of order n. Section 8.8, A Differential Equation with Nonconstant Coefficients. page 567
Problem number : Problem 23
Date solved : Tuesday, September 30, 2025 at 06:57:21 AM
CAS classification : [[_Emden, _Fowler], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} t^{2} y^{\prime \prime }+t y^{\prime }+25 y&=0 \end{align*}

With initial conditions

\begin{align*} y \left (1\right )&=\frac {3 \sqrt {3}}{2} \\ y^{\prime }\left (1\right )&={\frac {15}{2}} \\ \end{align*}

✓ Maple. Time used: 0.089 (sec). Leaf size: 22

ode:=t^2*diff(diff(y(t),t),t)+t*diff(y(t),t)+25*y(t) = 0; 
ic:=[y(1) = 3/2*3^(1/2), D(y)(1) = 15/2]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

\[ y = \frac {3 \sin \left (5 \ln \left (t \right )\right )}{2}+\frac {3 \sqrt {3}\, \cos \left (5 \ln \left (t \right )\right )}{2} \]

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 26

ode=t^2*D[y[t],{t,2}]+t*D[y[t],t]+25*y[t]==0; 
ic={y[1]==3*Sqrt[3]/2,Derivative[1][y][1]==15/2}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {3}{2} \left (\sin (5 \log (t))+\sqrt {3} \cos (5 \log (t))\right ) \end{align*}

✓ Sympy. Time used: 0.124 (sec). Leaf size: 27

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(t**2*Derivative(y(t), (t, 2)) + t*Derivative(y(t), t) + 25*y(t),0) 
ics = {y(1): 3*sqrt(3)/2, Subs(Derivative(y(t), t), t, 1): 15/2} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \frac {3 \sin {\left (5 \log {\left (t \right )} \right )}}{2} + \frac {3 \sqrt {3} \cos {\left (5 \log {\left (t \right )} \right )}}{2} \]