23.3.444 problem 449

Internal problem ID [6158]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 3. THE DIFFERENTIAL EQUATION IS LINEAR AND OF SECOND ORDER, page 311
Problem number : 449
Date solved : Tuesday, September 30, 2025 at 02:23:52 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y+4 x^{2} y^{\prime \prime }&=\sqrt {x} \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 20

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

\[ y = \left (c_2 +\ln \left (x \right ) c_1 +\frac {\ln \left (x \right )^{2}}{8}\right ) \sqrt {x} \]

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 29

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

\begin{align*} y(x)&\to \frac {1}{8} \sqrt {x} \left (\log ^2(x)+4 c_2 \log (x)+8 c_1\right ) \end{align*}

✓ Sympy. Time used: 0.127 (sec). Leaf size: 20

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

\[ y{\left (x \right )} = \sqrt {x} \left (C_{1} + C_{2} \log {\left (x \right )} + \frac {\log {\left (x \right )}^{2}}{8}\right ) \]