73.9.26 problem 14.3 (b)

Internal problem ID [15287]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 14. Higher order equations and the reduction of order method. Additional exercises page 277
Problem number : 14.3 (b)
Date solved : Tuesday, January 28, 2025 at 07:51:35 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-6 y^{\prime }+8 y&={\mathrm e}^{4 x} \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: 23

dsolve([diff(y(x),x$2)-6*diff(y(x),x)+8*y(x)=exp(4*x),exp(2*x)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.024 (sec). Leaf size: 31

DSolve[D[y[x],{x,2}]-6*D[y[x],x]+8*y[x]==Exp[4*x],y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to c_1 e^{2 x}+e^{4 x} \left (\frac {x}{2}-\frac {1}{4}+c_2\right ) \]