75.16.51 problem 524

Internal problem ID [16953]
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 : 524
Date solved : Monday, March 31, 2025 at 03:36:17 PM
CAS classification : [[_2nd_order, _missing_y]]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 25

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

\[ y = \frac {\left (-9 x^{2}-6 c_1 -6 x -2\right ) {\mathrm e}^{-3 x}}{18}+c_2 \]

✓ Mathematica. Time used: 2.529 (sec). Leaf size: 69

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

\begin{align*} y(x)\to \int _1^x\frac {1}{2} e^{-3 K[1]} \left (3 K[1]^2+2 c_1\right )dK[1]+c_2 \\ y(x)\to -\frac {1}{18} e^{-3 x} \left (9 x^2+6 x+2\right )+\frac {17}{18 e^3}+c_2 \\ \end{align*}

✓ Sympy. Time used: 0.246 (sec). Leaf size: 19

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

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