73.9.16 problem 14.2 (f)

Internal problem ID [15277]
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 : 14.2 (f)
Date solved : Tuesday, January 28, 2025 at 07:51:28 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-\left (4+\frac {2}{x}\right ) y^{\prime }+\left (4+\frac {4}{x}\right ) y&=0 \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: 16

dsolve([diff(y(x),x$2)-(4+2/x)*diff(y(x),x)+(4+4/x)*y(x)=0,exp(2*x)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.026 (sec). Leaf size: 25

DSolve[D[y[x],{x,2}]-(4+2/x)*D[y[x],x]+(4+4/x)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {1}{3} e^{2 x} \left (c_2 x^3+3 c_1\right ) \]