69.18.2 problem 591

Internal problem ID [18377]
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 : 591
Date solved : Thursday, October 02, 2025 at 03:11:07 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-6 y^{\prime }+9 y&=9 x^{2}-12 x +2 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=1 \\ y^{\prime }\left (0\right )&=3 \\ \end{align*}

✓ Maple. Time used: 0.034 (sec). Leaf size: 12

ode:=diff(diff(y(x),x),x)-6*diff(y(x),x)+9*y(x) = 9*x^2-12*x+2; 
ic:=[y(0) = 1, D(y)(0) = 3]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 14

ode=D[y[x],{x,2}]-6*D[y[x],x]+9*y[x]==9*x^2-12*x+2; 
ic={y[0]==1,Derivative[1][y][0] ==3}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.115 (sec). Leaf size: 10

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-9*x**2 + 12*x + 9*y(x) - 6*Derivative(y(x), x) + Derivative(y(x), (x, 2)) - 2,0) 
ics = {y(0): 1, Subs(Derivative(y(x), x), x, 0): 3} 
dsolve(ode,func=y(x),ics=ics)
 

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