12.10.21 problem 21

Internal problem ID [1825]
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 : 21
Date solved : Saturday, March 29, 2025 at 11:40:42 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} 4 x^{2} y^{\prime \prime }-4 x y^{\prime }+\left (4 x^{2}+3\right ) y&=x^{{7}/{2}} \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 21

ode:=4*x^2*diff(diff(y(x),x),x)-4*x*diff(y(x),x)+(4*x^2+3)*y(x) = x^(7/2); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\sqrt {x}\, \left (4 \cos \left (x \right ) c_1 +4 \sin \left (x \right ) c_2 +x \right )}{4} \]

✓ Mathematica. Time used: 0.076 (sec). Leaf size: 48

ode=4*x^2*D[y[x],{x,2}]-4*x*D[y[x],x]+(4*x^2+3)*y[x]==x^(7/2); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {1}{4} e^{-i x} \sqrt {x} \left (e^{i x} x-2 i c_2 e^{2 i x}+4 c_1\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x**(7/2) + 4*x**2*Derivative(y(x), (x, 2)) - 4*x*Derivative(y(x), x) + (4*x**2 + 3)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE x**(5/2)/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