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 : Monday, January 27, 2025 at 08:02:23 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*}

✓ Solution by Maple

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

dsolve([diff(y(x),x$2)-4*diff(y(x),x)+4*y(x)=4*exp(2*x)*ln(x),exp(2*x)],singsol=all)
 

\[ y \left (x \right ) = {\mathrm e}^{2 x} \left (2 \ln \left (x \right ) x^{2}+c_{1} x -3 x^{2}+c_{2} \right ) \]

✓ Solution by Mathematica

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

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

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