73.15.22 problem 22.8

Internal problem ID [15382]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 22. Method of undetermined coefficients. Additional exercises page 412
Problem number : 22.8
Date solved : Monday, March 31, 2025 at 01:36:06 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+9 y&=39 x \,{\mathrm e}^{2 x} \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.049 (sec). Leaf size: 26

ode:=diff(diff(y(x),x),x)+9*y(x) = 39*x*exp(2*x); 
ic:=y(0) = 1, D(y)(0) = 0; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = -\frac {5 \sin \left (3 x \right )}{13}+\frac {25 \cos \left (3 x \right )}{13}+3 \,{\mathrm e}^{2 x} \left (-\frac {4}{13}+x \right ) \]

✓ Mathematica. Time used: 0.022 (sec). Leaf size: 34

ode=D[y[x],{x,2}]+9*y[x]==39*x*Exp[2*x]; 
ic={y[0]==1,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {1}{13} \left (3 e^{2 x} (13 x-4)-5 \sin (3 x)+25 \cos (3 x)\right ) \]

✓ Sympy. Time used: 0.153 (sec). Leaf size: 36

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-39*x*exp(2*x) + 9*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): 1, Subs(Derivative(y(x), x), x, 0): 0} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = 3 x e^{2 x} - \frac {12 e^{2 x}}{13} - \frac {5 \sin {\left (3 x \right )}}{13} + \frac {25 \cos {\left (3 x \right )}}{13} \]