6.10.20 problem 20

Internal problem ID [1824]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 5 linear second order equations. Section 5.7 Variation of Parameters. Page 262
Problem number : 20
Date solved : Tuesday, September 30, 2025 at 05:20:06 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} 4 x^{2} y^{\prime \prime }-4 x y^{\prime }+\left (-16 x^{2}+3\right ) y&=8 x^{{5}/{2}} \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 22

ode:=4*x^2*diff(diff(y(x),x),x)-4*x*diff(y(x),x)+(-16*x^2+3)*y(x) = 8*x^(5/2); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \sqrt {x}\, \left (-\frac {1}{2}+c_2 \sinh \left (2 x \right )+c_1 \cosh \left (2 x \right )\right ) \]

✓ Mathematica. Time used: 0.024 (sec). Leaf size: 39

ode=4*x^2*D[y[x],{x,2}]-4*x*D[y[x],x]+(3-16*x^2)*y[x]==8*x^(5/2); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-8*x**(5/2) + 4*x**2*Derivative(y(x), (x, 2)) - 4*x*Derivative(y(x), x) + (3 - 16*x**2)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE 2*x**(3/2) + 4*x*y(x) - x*Derivative(y(x), (x, 2)) + Derivative(y(x), x) - 3*y(x)/(4*x) cannot be solved by the factorable group method