90.20.10 problem 10

Internal problem ID [25305]
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 : 10
Date solved : Friday, October 03, 2025 at 08:08:55 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }+\sin \left (t \right ) y&=0 \end{align*}

✓ Maple. Time used: 0.277 (sec). Leaf size: 34

ode:=diff(diff(y(t),t),t)+2*diff(y(t),t)+sin(t)*y(t) = 0; 
dsolve(ode,y(t), singsol=all);
 

\[ y = {\mathrm e}^{-t} \left (c_2 \operatorname {MathieuS}\left (-4, -2, -\frac {\pi }{4}+\frac {t}{2}\right )+c_1 \operatorname {MathieuC}\left (-4, -2, -\frac {\pi }{4}+\frac {t}{2}\right )\right ) \]

✓ Mathematica. Time used: 0.02 (sec). Leaf size: 44

ode=D[y[t],{t,2}]+2*D[y[t],t]+Sin[t]*y[t]==0; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to e^{-t} \left (c_1 \text {MathieuC}\left [-4,-2,\frac {1}{4} (\pi -2 t)\right ]+c_2 \text {MathieuS}\left [-4,-2,\frac {1}{4} (2 t-\pi )\right ]\right ) \end{align*}

✗ Sympy

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

False