20.11.12 problem Problem 15

Internal problem ID [3794]
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 15
Date solved : Sunday, March 30, 2025 at 02:08:42 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} 4 x^{2} y^{\prime \prime }+y&=\sqrt {x}\, \ln \left (x \right ) \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&=\sqrt {x} \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 20

ode:=4*x^2*diff(diff(y(x),x),x)+y(x) = x^(1/2)*ln(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (c_2 +\ln \left (x \right ) c_1 +\frac {\ln \left (x \right )^{3}}{24}\right ) \sqrt {x} \]

✓ Mathematica. Time used: 0.025 (sec). Leaf size: 29

ode=4*x^2*D[y[x],{x,2}]+y[x]==Sqrt[x]*Log[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {1}{24} \sqrt {x} \left (\log ^3(x)+12 c_2 \log (x)+24 c_1\right ) \]

✓ Sympy. Time used: 0.219 (sec). Leaf size: 20

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-sqrt(x)*log(x) + 4*x**2*Derivative(y(x), (x, 2)) + y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \sqrt {x} \left (C_{1} + C_{2} \log {\left (x \right )} + \frac {\log {\left (x \right )}^{3}}{24}\right ) \]