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 : Tuesday, March 04, 2025 at 12:29:10 PM
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*}

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

ode:=x^2*diff(diff(y(x),x),x)-(x-3/16)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

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

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

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

✓ Sympy. Time used: 0.087 (sec). Leaf size: 34

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) - (x - 3/16)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \sqrt {x} \left (C_{1} J_{\frac {1}{2}}\left (2 i \sqrt {x}\right ) + C_{2} Y_{\frac {1}{2}}\left (2 i \sqrt {x}\right )\right ) \]