74.12.48 problem 56

Internal problem ID [16276]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.4, page 163
Problem number : 56
Date solved : Thursday, March 13, 2025 at 08:09:23 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 22

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

\[ y = \sin \left (2 \ln \left (t \right )\right ) c_{2} +\cos \left (2 \ln \left (t \right )\right ) c_{1} +\frac {t}{5} \]

✓ Mathematica. Time used: 0.057 (sec). Leaf size: 68

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

\[ y(t)\to \cos (2 \log (t)) \int _1^t-\frac {1}{2} \sin (2 \log (K[1]))dK[1]+\sin (2 \log (t)) \int _1^t\frac {1}{2} \cos (2 \log (K[2]))dK[2]+c_1 \cos (2 \log (t))+c_2 \sin (2 \log (t)) \]

✓ Sympy. Time used: 0.246 (sec). Leaf size: 22

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

\[ y{\left (t \right )} = C_{1} \sin {\left (2 \log {\left (t \right )} \right )} + C_{2} \cos {\left (2 \log {\left (t \right )} \right )} + \frac {t}{5} \]