34.6.12 problem 12

Internal problem ID [6086]
Book : A treatise on ordinary and partial differential equations by William Woolsey Johnson. 1913
Section : Chapter IX, Special forms of differential equations. Examples XVII. page 247
Problem number : 12
Date solved : Sunday, March 30, 2025 at 10:38:11 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {25}{4}\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.048 (sec). Leaf size: 43

ode:=x^2*diff(diff(y(x),x),x)+x*diff(y(x),x)+(x^2-25/4)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {-3 c_2 \left (i x -\frac {1}{3} x^{2}+1\right ) {\mathrm e}^{-i x}+3 \left (i x +\frac {1}{3} x^{2}-1\right ) c_1 \,{\mathrm e}^{i x}}{x^{{5}/{2}}} \]

✓ Mathematica. Time used: 0.071 (sec). Leaf size: 59

ode=x^2*D[y[x],{x,2}]+x*D[y[x],x]+(x^2-25/4)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to -\frac {\sqrt {\frac {2}{\pi }} \left (\left (-c_2 x^2+3 c_1 x+3 c_2\right ) \cos (x)+\left (c_1 \left (x^2-3\right )+3 c_2 x\right ) \sin (x)\right )}{x^{5/2}} \]

✓ Sympy. Time used: 0.232 (sec). Leaf size: 15

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + x*Derivative(y(x), x) + (x**2 - 25/4)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{1} J_{\frac {5}{2}}\left (x\right ) + C_{2} Y_{\frac {5}{2}}\left (x\right ) \]