76.33.13 problem Ex. 13

Internal problem ID [20191]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter VI. Linear equations with constant coefficients. Examples on chapter VI, page 80
Problem number : Ex. 13
Date solved : Thursday, October 02, 2025 at 05:34:05 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y-2 y^{\prime }+y^{\prime \prime }&=x^{2} {\mathrm e}^{3 x} \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 29

ode:=diff(diff(y(x),x),x)-2*diff(y(x),x)+y(x) = x^2*exp(3*x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (\left (x^{2}-2 x +\frac {3}{2}\right ) {\mathrm e}^{2 x}+4 c_1 x +4 c_2 \right ) {\mathrm e}^{x}}{4} \]

✓ Mathematica. Time used: 0.024 (sec). Leaf size: 36

ode=D[y[x],{x,2}]-2*D[y[x],x]+y[x]==x^2*Exp[3*x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{8} e^{3 x} \left (2 x^2-4 x+3\right )+e^x (c_2 x+c_1) \end{align*}

✓ Sympy. Time used: 0.176 (sec). Leaf size: 31

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

\[ y{\left (x \right )} = \left (C_{1} + x \left (C_{2} - \frac {e^{2 x}}{2}\right ) + \frac {\left (2 x^{2} + 3\right ) e^{2 x}}{8}\right ) e^{x} \]