58.11.23 problem 23

Internal problem ID [14754]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 4, Section 4.3. The method of undetermined coefficients. Exercises page 151
Problem number : 23
Date solved : Thursday, October 02, 2025 at 09:50:48 AM
CAS classification : [[_high_order, _missing_y]]

\begin{align*} y^{\prime \prime \prime \prime }+2 y^{\prime \prime \prime }-3 y^{\prime \prime }&=18 x^{2}+16 x \,{\mathrm e}^{x}+4 \,{\mathrm e}^{3 x}-9 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 54

ode:=diff(diff(diff(diff(y(x),x),x),x),x)+2*diff(diff(diff(y(x),x),x),x)-3*diff(diff(y(x),x),x) = 18*x^2+16*x*exp(x)+4*exp(3*x)-9; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {{\mathrm e}^{-3 x} c_2}{9}+\frac {{\mathrm e}^{3 x}}{27}+\frac {\left (8 x^{2}+4 c_1 -36 x +57\right ) {\mathrm e}^{x}}{4}-\frac {x^{4}}{2}-\frac {4 x^{3}}{3}-\frac {19 x^{2}}{6}+c_3 x +c_4 \]

✓ Mathematica. Time used: 1.032 (sec). Leaf size: 70

ode=D[y[x],{x,4}]+2*D[y[x],{x,3}]-3*D[y[x],{x,2}]==18*x^2+16*x*Exp[x]+4*Exp[3*x]-9; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {1}{6} \left (3 x^2+8 x+19\right ) x^2+\frac {1}{4} e^x \left (8 x^2-36 x+57+4 c_2\right )+\frac {e^{3 x}}{27}+c_4 x+\frac {1}{9} c_1 e^{-3 x}+c_3 \end{align*}

✓ Sympy. Time used: 0.130 (sec). Leaf size: 54

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-18*x**2 - 16*x*exp(x) - 4*exp(3*x) - 3*Derivative(y(x), (x, 2)) + 2*Derivative(y(x), (x, 3)) + Derivative(y(x), (x, 4)) + 9,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{1} + C_{3} e^{- 3 x} + C_{4} e^{x} - \frac {x^{4}}{2} - \frac {4 x^{3}}{3} + x^{2} \left (2 e^{x} - \frac {19}{6}\right ) + x \left (C_{2} - 9 e^{x}\right ) + \frac {e^{3 x}}{27} \]