50.3.30 problem 30

Internal problem ID [10174]
Book : Own collection of miscellaneous problems
Section : section 3.0
Problem number : 30
Date solved : Tuesday, September 30, 2025 at 07:11:04 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} 4 x^{2} y^{\prime \prime }+y&=8 \sqrt {x}\, \left (1+\ln \left (x \right )\right ) \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 24

ode:=4*x^2*diff(diff(y(x),x),x)+y(x) = 8*x^(1/2)*(ln(x)+1); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (c_2 +\ln \left (x \right ) c_1 +\frac {\ln \left (x \right )^{3}}{3}+\ln \left (x \right )^{2}\right ) \sqrt {x} \]

✓ Mathematica. Time used: 0.022 (sec). Leaf size: 37

ode=4*x^2*D[y[x],{x,2}]+y[x] == 8*Sqrt[x]*(1+Log[x]); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{6} \sqrt {x} \left (2 \log ^3(x)+6 \log ^2(x)+3 c_2 \log (x)+6 c_1\right ) \end{align*}

✓ Sympy. Time used: 0.163 (sec). Leaf size: 26

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-8*sqrt(x)*(log(x) + 1) + 4*x**2*Derivative(y(x), (x, 2)) + y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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