69.18.5 problem 594

Internal problem ID [18380]
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 : 594
Date solved : Thursday, October 02, 2025 at 03:11:09 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} 6 y-5 y^{\prime }+y^{\prime \prime }&=\left (12 x -7\right ) {\mathrm e}^{-x} \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.028 (sec). Leaf size: 21

ode:=diff(diff(y(x),x),x)-5*diff(y(x),x)+6*y(x) = (12*x-7)*exp(-x); 
ic:=[y(0) = 0, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 25

ode=D[y[x],{x,2}]-5*D[y[x],x]+6*y[x]==(12*x-7)*Exp[-x]; 
ic={y[0]==0,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.149 (sec). Leaf size: 17

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((7 - 12*x)*exp(-x) + 6*y(x) - 5*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(x), x), x, 0): 0} 
dsolve(ode,func=y(x),ics=ics)
 

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