20.9.27 problem Problem 27

Internal problem ID [3771]
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 27
Date solved : Tuesday, March 04, 2025 at 05:15:07 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-4 y^{\prime }+4 y&=5 x \,{\mathrm e}^{2 x} \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=1\\ y^{\prime }\left (0\right )&=0 \end{align*}

✓ Maple. Time used: 0.018 (sec). Leaf size: 20

ode:=diff(diff(y(x),x),x)-4*diff(y(x),x)+4*y(x) = 5*x*exp(2*x); 
ic:=y(0) = 1, D(y)(0) = 0; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y \left (x \right ) = \frac {{\mathrm e}^{2 x} \left (5 x^{3}-12 x +6\right )}{6} \]

✓ Mathematica. Time used: 0.024 (sec). Leaf size: 24

ode=D[y[x],{x,2}]-4*D[y[x],x]+4*y[x]==5*x*Exp[2*x]; 
ic={y[0]==1,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {1}{6} e^{2 x} \left (5 x^3-12 x+6\right ) \]

✓ Sympy. Time used: 0.280 (sec). Leaf size: 19

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-5*x*exp(2*x) + 4*y(x) - 4*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): 1, Subs(Derivative(y(x), x), x, 0): 0} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (x \left (\frac {5 x^{2}}{6} - 2\right ) + 1\right ) e^{2 x} \]