58.11.27 problem 27

Internal problem ID [14758]
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 : 27
Date solved : Thursday, October 02, 2025 at 09:50:51 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-8 y^{\prime }+15 y&=9 x \,{\mathrm e}^{2 x} \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=5 \\ y^{\prime }\left (0\right )&=10 \\ \end{align*}

✓ Maple. Time used: 0.034 (sec). Leaf size: 24

ode:=diff(diff(y(x),x),x)-8*diff(y(x),x)+15*y(x) = 9*x*exp(2*x); 
ic:=[y(0) = 5, D(y)(0) = 10]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 28

ode=D[y[x],{x,2}]-8*D[y[x],x]+15*y[x]==9*x*Exp[2*x]; 
ic={y[0]==5,Derivative[1][y][0] ==10}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.175 (sec). Leaf size: 24

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-9*x*exp(2*x) + 15*y(x) - 8*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): 5, Subs(Derivative(y(x), x), x, 0): 10} 
dsolve(ode,func=y(x),ics=ics)
 

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