75.16.54 problem 527

Internal problem ID [16956]
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. Trial and error method. Exercises page 132
Problem number : 527
Date solved : Monday, March 31, 2025 at 03:36:22 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y^{\prime }+y&=\left (x^{2}+x \right ) {\mathrm e}^{x} \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 43

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

\[ y = {\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right ) c_1 +{\mathrm e}^{-\frac {x}{2}} \sin \left (\frac {\sqrt {3}\, x}{2}\right ) c_2 +\frac {{\mathrm e}^{x} \left (x^{2}-x +\frac {1}{3}\right )}{3} \]

✓ Mathematica. Time used: 0.031 (sec). Leaf size: 65

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

\[ y(x)\to \frac {1}{9} e^{-x/2} \left (e^{3 x/2} \left (3 x^2-3 x+1\right )+9 c_2 \cos \left (\frac {\sqrt {3} x}{2}\right )+9 c_1 \sin \left (\frac {\sqrt {3} x}{2}\right )\right ) \]

✓ Sympy. Time used: 0.287 (sec). Leaf size: 46

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

\[ y{\left (x \right )} = \left (C_{1} \sin {\left (\frac {\sqrt {3} x}{2} \right )} + C_{2} \cos {\left (\frac {\sqrt {3} x}{2} \right )}\right ) e^{- \frac {x}{2}} + \frac {\left (3 x^{2} - 3 x + 1\right ) e^{x}}{9} \]