8.11.23 problem 46

Internal problem ID [891]
Book : Differential equations and linear algebra, 3rd ed., Edwards and Penney
Section : Section 5.5, Nonhomogeneous equations and undetermined coefficients Page 351
Problem number : 46
Date solved : Saturday, March 29, 2025 at 10:33:36 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y&=x \cos \left (x \right )^{3} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 43

ode:=diff(diff(y(x),x),x)+y(x) = x*cos(x)^3; 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\frac {x \cos \left (x \right )^{3}}{8}+\frac {3 \sin \left (x \right ) \cos \left (x \right )^{2}}{32}+\frac {\left (9 x +32 c_1 \right ) \cos \left (x \right )}{32}+\frac {3 \left (x^{2}+\frac {16 c_2}{3}+\frac {3}{4}\right ) \sin \left (x \right )}{16} \]

✓ Mathematica. Time used: 0.05 (sec). Leaf size: 49

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

\[ y(x)\to \frac {1}{128} \left (\sin (x) \left (24 x^2+6 \cos (2 x)-9+128 c_2\right )-4 x \cos (3 x)+8 (3 x+16 c_1) \cos (x)\right ) \]

✓ Sympy. Time used: 0.667 (sec). Leaf size: 41

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

\[ y{\left (x \right )} = \left (C_{1} + \frac {3 x^{2}}{16}\right ) \sin {\left (x \right )} + \left (C_{2} + \frac {x \sin ^{2}{\left (x \right )}}{8} + \frac {5 x}{32}\right ) \cos {\left (x \right )} - \frac {3 \sin ^{3}{\left (x \right )}}{32} \]