89.16.29 problem 29

Internal problem ID [24679]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 9. Nonhomogeneous Equations: Undetermined coefficients. Exercises at page 140
Problem number : 29
Date solved : Thursday, October 02, 2025 at 10:46:59 PM
CAS classification : [[_2nd_order, _missing_y]]

\begin{align*} y^{\prime \prime }+3 y^{\prime }&=-18 x \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=0 \\ y^{\prime }\left (0\right )&=5 \\ \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 20

ode:=diff(diff(y(x),x),x)+3*diff(y(x),x) = -18*x; 
ic:=[y(0) = 0, D(y)(0) = 5]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = -{\mathrm e}^{-3 x}-3 x^{2}+2 x +1 \]

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 24

ode=D[y[x],{x,2}]+3*y[x]== -18*x; 
ic={y[0]==0,Derivative[1][y][0] ==5}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {11 \sin \left (\sqrt {3} x\right )}{\sqrt {3}}-6 x \end{align*}

✓ Sympy. Time used: 0.056 (sec). Leaf size: 22

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(18*x + 3*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(x), x), x, 0): 5} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = - 6 x + \frac {11 \sqrt {3} \sin {\left (\sqrt {3} x \right )}}{3} \]