75.16.64 problem 537

Internal problem ID [16966]
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 : 537
Date solved : Monday, March 31, 2025 at 03:36:42 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 23

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

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

✓ Mathematica. Time used: 0.04 (sec). Leaf size: 50

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

\[ y(x)\to \frac {1}{2} e^x \left (2 e^x \int _1^xe^{-K[1]} K[1]dK[1]-x^2+2 \left (c_2 e^x+c_1\right )\right ) \]

✓ Sympy. Time used: 0.207 (sec). Leaf size: 19

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

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