90.20.5 problem 5

Internal problem ID [25300]
Book : Ordinary Differential Equations. By William Adkins and Mark G Davidson. Springer. NY. 2010. ISBN 978-1-4614-3617-1
Section : Chapter 5. Second Order Linear Differential Equations. Exercises at page 337
Problem number : 5
Date solved : Thursday, October 02, 2025 at 11:59:41 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} 3 t^{2} y^{\prime \prime }+2 t y^{\prime }+y&={\mathrm e}^{2 t} \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 94

ode:=3*t^2*diff(diff(y(t),t),t)+2*t*diff(y(t),t)+y(t) = exp(2*t); 
dsolve(ode,y(t), singsol=all);
 

\[ y = \frac {t^{{1}/{6}} \left (2 \int \frac {\cos \left (\frac {\sqrt {11}\, \ln \left (t \right )}{6}\right ) {\mathrm e}^{2 t}}{t^{{7}/{6}}}d t \sqrt {11}\, \sin \left (\frac {\sqrt {11}\, \ln \left (t \right )}{6}\right )-2 \int \frac {\sin \left (\frac {\sqrt {11}\, \ln \left (t \right )}{6}\right ) {\mathrm e}^{2 t}}{t^{{7}/{6}}}d t \sqrt {11}\, \cos \left (\frac {\sqrt {11}\, \ln \left (t \right )}{6}\right )+11 c_2 \sin \left (\frac {\sqrt {11}\, \ln \left (t \right )}{6}\right )+11 c_1 \cos \left (\frac {\sqrt {11}\, \ln \left (t \right )}{6}\right )\right )}{11} \]

✓ Mathematica. Time used: 0.508 (sec). Leaf size: 252

ode=3*t^2*D[y[t],{t,2}]+2*t*D[y[t],{t,1}]+y[t]==Exp[2*t]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \frac {2^{\frac {1}{6} \left (1+i \sqrt {11}\right )} t^{1-\frac {i \sqrt {11}}{6}} (-t)^{\frac {1}{6} i \left (\sqrt {11}+5 i\right )} \Gamma \left (-\frac {1}{6}-\frac {i \sqrt {11}}{6},-2 t\right ) \left (\sin \left (\frac {1}{6} \sqrt {11} \log (t)\right )-i \cos \left (\frac {1}{6} \sqrt {11} \log (t)\right )\right )}{\sqrt {11}}-\frac {2^{\frac {1}{6} \left (1-i \sqrt {11}\right )} t^{\frac {i \sqrt {11}}{6}} (-t)^{\frac {1}{6}-\frac {i \sqrt {11}}{6}} \Gamma \left (\frac {1}{6} i \left (i+\sqrt {11}\right ),-2 t\right ) \left (\sin \left (\frac {1}{6} \sqrt {11} \log (t)\right )+i \cos \left (\frac {1}{6} \sqrt {11} \log (t)\right )\right )}{\sqrt {11}}+\sqrt [6]{t} \left (c_2 \cos \left (\frac {1}{6} \sqrt {11} \log (t)\right )+c_1 \sin \left (\frac {1}{6} \sqrt {11} \log (t)\right )\right ) \end{align*}

✗ Sympy

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(3*t**2*Derivative(y(t), (t, 2)) + 2*t*Derivative(y(t), t) + y(t) - exp(2*t),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

Timed Out