85.67.11 problem 11

Internal problem ID [22902]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter 4. Linear differential equations. A Exercises at page 216
Problem number : 11
Date solved : Thursday, October 02, 2025 at 09:16:26 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} 4 y+y^{\prime \prime }&=x \left (1+\cos \left (x \right )\right ) \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 29

ode:=diff(diff(y(x),x),x)+4*y(x) = x*(cos(x)+1); 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.151 (sec). Leaf size: 38

ode=D[y[x],{x,2}]+4*y[x]==x*(1+Cos[x]); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {x}{4}+\frac {2 \sin (x)}{9}+\frac {1}{3} x \cos (x)+c_1 \cos (2 x)+c_2 \sin (2 x) \end{align*}

✓ Sympy. Time used: 0.064 (sec). Leaf size: 32

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*(cos(x) + 1) + 4*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{1} \sin {\left (2 x \right )} + C_{2} \cos {\left (2 x \right )} + \frac {x \cos {\left (x \right )}}{3} + \frac {x}{4} + \frac {2 \sin {\left (x \right )}}{9} \]