69.16.8 problem 481

Internal problem ID [18268]
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 : 481
Date solved : Thursday, October 02, 2025 at 03:09:57 PM
CAS classification : [[_2nd_order, _missing_y]]

\begin{align*} y^{\prime \prime }-4 y^{\prime }&=x \,{\mathrm e}^{4 x} \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 25

ode:=diff(diff(y(x),x),x)-4*diff(y(x),x) = x*exp(4*x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (8 x^{2}+16 c_1 -4 x +1\right ) {\mathrm e}^{4 x}}{64}+c_2 \]

✓ Mathematica. Time used: 1.195 (sec). Leaf size: 67

ode=D[y[x],{x,2}]-4*D[y[x],x]==x*Exp[4*x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \int _1^x\frac {1}{2} e^{4 K[1]} \left (K[1]^2+2 c_1\right )dK[1]+c_2\\ y(x)&\to \frac {1}{64} e^{4 x} \left (8 x^2-4 x+1\right )-\frac {5 e^4}{64}+c_2 \end{align*}

✓ Sympy. Time used: 0.133 (sec). Leaf size: 19

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*exp(4*x) - 4*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{1} + \left (C_{2} + \frac {x^{2}}{8} - \frac {x}{16}\right ) e^{4 x} \]