20.9.18 problem Problem 18

Internal problem ID [3762]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 8, Linear differential equations of order n. Section 8.7, The Variation of Parameters Method. page 556
Problem number : Problem 18
Date solved : Sunday, March 30, 2025 at 02:07:39 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }+4 y&=15 \,{\mathrm e}^{-2 x} \ln \left (x \right )+25 \cos \left (x \right ) \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 39

ode:=diff(diff(y(x),x),x)+4*diff(y(x),x)+4*y(x) = 15*exp(-2*x)*ln(x)+25*cos(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (30 \ln \left (x \right ) x^{2}-45 x^{2}+4 c_1 x +4 c_2 \right ) {\mathrm e}^{-2 x}}{4}+3 \cos \left (x \right )+4 \sin \left (x \right ) \]

✓ Mathematica. Time used: 0.193 (sec). Leaf size: 54

ode=D[y[x],{x,2}]+4*D[y[x],x]+4*y[x]==15*Exp[-2*x]*Log[x]+25*Cos[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {1}{4} e^{-2 x} \left (-45 x^2+30 x^2 \log (x)+16 e^{2 x} \sin (x)+12 e^{2 x} \cos (x)+4 c_2 x+4 c_1\right ) \]

✓ Sympy. Time used: 0.824 (sec). Leaf size: 36

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*y(x) - 25*cos(x) + 4*Derivative(y(x), x) + Derivative(y(x), (x, 2)) - 15*exp(-2*x)*log(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (C_{1} + x \left (C_{2} + \frac {15 x \log {\left (x \right )}}{2} - \frac {45 x}{4}\right )\right ) e^{- 2 x} + 4 \sin {\left (x \right )} + 3 \cos {\left (x \right )} \]