89.23.26 problem 30

Internal problem ID [24830]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 10. Nonhomogeneous Equations: Operational methods. Miscellaneous Exercises at page 162
Problem number : 30
Date solved : Thursday, October 02, 2025 at 10:48:24 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 y+y^{\prime \prime }&=2 x -8 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=1 \\ y^{\prime }\left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.012 (sec). Leaf size: 21

ode:=diff(diff(y(x),x),x)+4*y(x) = 2*x-8; 
ic:=[y(0) = 1, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 24

ode=D[y[x],{x,2}]+4*y[x]== 2*x-8; 
ic={y[0]==1,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{2} (x+6 \cos (2 x)-\sin (x) \cos (x)-4) \end{align*}

✓ Sympy. Time used: 0.053 (sec). Leaf size: 17

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-2*x + y(x) + Derivative(y(x), (x, 2)) + 8,0) 
ics = {y(0): 1, Subs(Derivative(y(x), x), x, 0): 0} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = 2 x - 2 \sin {\left (x \right )} + 9 \cos {\left (x \right )} - 8 \]