20.11.11 problem Problem 14

Internal problem ID [3793]
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 8.9, Reduction of Order. page 572
Problem number : Problem 14
Date solved : Sunday, March 30, 2025 at 02:08:40 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using reduction of order method given that one solution is

\begin{align*} y&={\mathrm e}^{2 x} \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 26

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

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

✓ Mathematica. Time used: 0.024 (sec). Leaf size: 30

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

\[ y(x)\to e^{2 x} \left (-3 x^2+2 x^2 \log (x)+c_2 x+c_1\right ) \]

✓ Sympy. Time used: 0.262 (sec). Leaf size: 22

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

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