69.18.12 problem 601

Internal problem ID [18387]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 2 (Higher order ODEs). Section 15.3 Nonhomogeneous linear equations with constant coefficients. Initial value problem. Exercises page 140
Problem number : 601
Date solved : Thursday, October 02, 2025 at 03:11:15 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-6 y^{\prime }+9 y&=16 \,{\mathrm e}^{-x}+9 x -6 \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.023 (sec). Leaf size: 16

ode:=diff(diff(y(x),x),x)-6*diff(y(x),x)+9*y(x) = 16*exp(-x)+9*x-6; 
ic:=[y(0) = 1, D(y)(0) = 1]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = {\mathrm e}^{3 x} x +x +{\mathrm e}^{-x} \]

✓ Mathematica. Time used: 0.256 (sec). Leaf size: 137

ode=D[y[x],{x,2}]-6*D[y[x],x]+9*y[x]==16*Exp[-x]+9*x-6; 
ic={y[0]==1,Derivative[1][y][0] ==1}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to e^{3 x} \left (x \left (-\int _1^0e^{-4 K[2]} \left (e^{K[2]} (9 K[2]-6)+16\right )dK[2]\right )+x \int _1^xe^{-4 K[2]} \left (e^{K[2]} (9 K[2]-6)+16\right )dK[2]+\int _1^x-e^{-4 K[1]} K[1] \left (e^{K[1]} (9 K[1]-6)+16\right )dK[1]-\int _1^0-e^{-4 K[1]} K[1] \left (e^{K[1]} (9 K[1]-6)+16\right )dK[1]-2 x+1\right ) \end{align*}

✓ Sympy. Time used: 0.150 (sec). Leaf size: 15

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-9*x + 9*y(x) - 6*Derivative(y(x), x) + Derivative(y(x), (x, 2)) + 6 - 16*exp(-x),0) 
ics = {y(0): 1, Subs(Derivative(y(x), x), x, 0): 1} 
dsolve(ode,func=y(x),ics=ics)
 

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