85.64.7 problem 1 (g)

Internal problem ID [22874]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter 4. Linear differential equations. A Exercises at page 213
Problem number : 1 (g)
Date solved : Thursday, October 02, 2025 at 09:16:04 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} t^{2} i^{\prime \prime }+2 i^{\prime } t +i&=t \ln \left (t \right ) \end{align*}

✓ Maple. Time used: 0.015 (sec). Leaf size: 38

ode:=t^2*diff(diff(i(t),t),t)+2*t*diff(i(t),t)+i(t) = t*ln(t); 
dsolve(ode,i(t), singsol=all);
 

\[ i = \frac {\sin \left (\frac {\sqrt {3}\, \ln \left (t \right )}{2}\right ) c_2}{\sqrt {t}}+\frac {\cos \left (\frac {\sqrt {3}\, \ln \left (t \right )}{2}\right ) c_1}{\sqrt {t}}+\frac {t \left (\ln \left (t \right )-1\right )}{3} \]

✓ Mathematica. Time used: 0.177 (sec). Leaf size: 55

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

\begin{align*} i(t)&\to \frac {1}{3} t (\log (t)-1)+\frac {c_2 \cos \left (\frac {1}{2} \sqrt {3} \log (t)\right )}{\sqrt {t}}+\frac {c_1 \sin \left (\frac {1}{2} \sqrt {3} \log (t)\right )}{\sqrt {t}} \end{align*}

✓ Sympy. Time used: 0.301 (sec). Leaf size: 46

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

\[ i{\left (t \right )} = \frac {C_{1} \sin {\left (\frac {\sqrt {3} \log {\left (t \right )}}{2} \right )} + C_{2} \cos {\left (\frac {\sqrt {3} \log {\left (t \right )}}{2} \right )} + \frac {t^{\frac {3}{2}} \left (\log {\left (t \right )} - 1\right )}{3}}{\sqrt {t}} \]