20.7.18 problem Problem 47

Internal problem ID [3733]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 8, Linear differential equations of order n. Section 8.3, The Method of Undetermined Coefficients. page 525
Problem number : Problem 47
Date solved : Sunday, March 30, 2025 at 02:06:48 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+6 y&=\sin \left (x \right )^{2} \cos \left (x \right )^{2} \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 28

ode:=diff(diff(y(x),x),x)+6*y(x) = sin(x)^2*cos(x)^2; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \sin \left (\sqrt {6}\, x \right ) c_2 +\cos \left (\sqrt {6}\, x \right ) c_1 +\frac {1}{48}+\frac {\cos \left (4 x \right )}{80} \]

✓ Mathematica. Time used: 0.753 (sec). Leaf size: 39

ode=D[y[x],{x,2}]+6*y[x]==Sin[x]^2*Cos[x]^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {1}{80} \cos (4 x)+c_1 \cos \left (\sqrt {6} x\right )+c_2 \sin \left (\sqrt {6} x\right )+\frac {1}{48} \]

✓ Sympy. Time used: 0.933 (sec). Leaf size: 39

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(6*y(x) - sin(x)**2*cos(x)**2 + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{1} \sin {\left (\sqrt {6} x \right )} + C_{2} \cos {\left (\sqrt {6} x \right )} + \frac {\sin ^{4}{\left (x \right )}}{10} - \frac {\sin ^{2}{\left (x \right )}}{10} + \frac {1}{30} \]