7.11.5 problem 5

Internal problem ID [326]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 2. Linear Equations of Higher Order. Section 2.5 (Nonhomogeneous equations and undetermined coefficients). Problems at page 161
Problem number : 5
Date solved : Tuesday, March 04, 2025 at 11:08:34 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 44

ode:=diff(diff(y(x),x),x)+diff(y(x),x)+y(x) = sin(x)^2; 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{-\frac {x}{2}} \sin \left (\frac {\sqrt {3}\, x}{2}\right ) c_2 +{\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right ) c_1 -\frac {\sin \left (2 x \right )}{13}+\frac {3 \cos \left (2 x \right )}{26}+\frac {1}{2} \]

✓ Mathematica. Time used: 1.653 (sec). Leaf size: 67

ode=D[y[x],{x,2}]+D[y[x],x]+y[x]==Sin[x]^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to -\frac {1}{13} \sin (2 x)+\frac {3}{26} \cos (2 x)+c_2 e^{-x/2} \cos \left (\frac {\sqrt {3} x}{2}\right )+c_1 e^{-x/2} \sin \left (\frac {\sqrt {3} x}{2}\right )+\frac {1}{2} \]

✓ Sympy. Time used: 4.867 (sec). Leaf size: 49

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

\[ y{\left (x \right )} = \left (C_{1} \sin {\left (\frac {\sqrt {3} x}{2} \right )} + C_{2} \cos {\left (\frac {\sqrt {3} x}{2} \right )}\right ) e^{- \frac {x}{2}} - \frac {\sin {\left (2 x \right )}}{13} + \frac {3 \cos {\left (2 x \right )}}{26} + \frac {1}{2} \]