67.15.5 problem 22.2

Internal problem ID [16723]
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 : 22.2
Date solved : Thursday, October 02, 2025 at 01:38:11 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-3 y^{\prime }-10 y&=-5 \,{\mathrm e}^{3 x} \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=5 \\ y^{\prime }\left (0\right )&=3 \\ \end{align*}

✓ Maple. Time used: 0.027 (sec). Leaf size: 23

ode:=diff(diff(y(x),x),x)-3*diff(y(x),x)-10*y(x) = -5*exp(3*x); 
ic:=[y(0) = 5, D(y)(0) = 3]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = 3 \,{\mathrm e}^{-2 x}+\frac {3 \,{\mathrm e}^{5 x}}{2}+\frac {{\mathrm e}^{3 x}}{2} \]

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 28

ode=D[y[x],{x,2}]-3*D[y[x],x]-10*y[x]==-5*Exp[3*x]; 
ic={y[0]==5,Derivative[1][y][0] ==3}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{2} e^{-2 x} \left (e^{5 x}+3 e^{7 x}+6\right ) \end{align*}

✓ Sympy. Time used: 0.139 (sec). Leaf size: 24

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-10*y(x) + 5*exp(3*x) - 3*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): 5, Subs(Derivative(y(x), x), x, 0): 3} 
dsolve(ode,func=y(x),ics=ics)
 

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