10.9.23 problem 29

Internal problem ID [1325]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Chapter 3, Second order linear equations, 3.4 Repeated roots, reduction of order , page 172
Problem number : 29
Date solved : Monday, January 27, 2025 at 04:51:02 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-\left (x -\frac {3}{16}\right ) y&=0 \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&=x^{{1}/{4}} {\mathrm e}^{2 \sqrt {x}} \end{align*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 25

dsolve([x^2*diff(y(x),x$2)-(x-1875/10000)*y(x)=0,x^(1/4)*exp(2*sqrt(x))],singsol=all)
 

\[ y = x^{{1}/{4}} \left (c_1 \sinh \left (2 \sqrt {x}\right )+c_2 \cosh \left (2 \sqrt {x}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.045 (sec). Leaf size: 41

DSolve[x^2*D[y[x],{x,2}]-(x-1875/10000)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

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