23.3.448 problem 453

Internal problem ID [6162]
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 : 453
Date solved : Tuesday, September 30, 2025 at 02:23:55 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} -\left (4 x^{2}+1\right ) y+4 x y^{\prime }+4 x^{2} y^{\prime \prime }&=4 \,{\mathrm e}^{x} x^{{3}/{2}} \end{align*}

✓ Maple. Time used: 0.015 (sec). Leaf size: 24

ode:=-(4*x^2+1)*y(x)+4*x*diff(y(x),x)+4*x^2*diff(diff(y(x),x),x) = 4*exp(x)*x^(3/2); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {{\mathrm e}^{x} x +2 \cosh \left (x \right ) c_1 +2 \sinh \left (x \right ) c_2}{2 \sqrt {x}} \]

✓ Mathematica. Time used: 0.025 (sec). Leaf size: 39

ode=-((1 + 4*x^2)*y[x]) + 4*x*D[y[x],x] + 4*x^2*D[y[x],{x,2}] == 4*E^x*x^(3/2); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-4*x**(3/2)*exp(x) + 4*x**2*Derivative(y(x), (x, 2)) + 4*x*Derivative(y(x), x) + (-4*x**2 - 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -sqrt(x)*exp(x) - x*y(x) + x*Derivative(y(x), (x, 2)) + Derivati