58.11.34 problem 34

Internal problem ID [14765]
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 : 34
Date solved : Thursday, October 02, 2025 at 09:50:56 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-y^{\prime }-6 y&=8 \,{\mathrm e}^{2 x}-5 \,{\mathrm e}^{3 x} \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.025 (sec). Leaf size: 25

ode:=diff(diff(y(x),x),x)-diff(y(x),x)-6*y(x) = 8*exp(2*x)-5*exp(3*x); 
ic:=[y(0) = 3, D(y)(0) = 5]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = {\mathrm e}^{-2 x}-2 \,{\mathrm e}^{2 x}+{\mathrm e}^{3 x} \left (4-x \right ) \]

✓ Mathematica. Time used: 0.026 (sec). Leaf size: 28

ode=D[y[x],{x,2}]-D[y[x],x]-6*y[x]==8*Exp[2*x]-5*Exp[3*x]; 
ic={y[0]==3,Derivative[1][y][0] ==5}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -e^{3 x} (x-4)+e^{-2 x}-2 e^{2 x} \end{align*}

✓ Sympy. Time used: 0.163 (sec). Leaf size: 24

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-6*y(x) + 5*exp(3*x) - 8*exp(2*x) - Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): 3, Subs(Derivative(y(x), x), x, 0): 5} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (4 - x\right ) e^{3 x} - 2 e^{2 x} + e^{- 2 x} \]