64.15.22 problem 22

Internal problem ID [13528]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 6, Series solutions of linear differential equations. Section 6.2 (Frobenius). Exercises page 251
Problem number : 22
Date solved : Monday, March 31, 2025 at 08:00:28 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+x^{2} y^{\prime }-\frac {3 y}{4}&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Maple. Time used: 0.024 (sec). Leaf size: 62

Order:=6; 
ode:=x^2*diff(diff(y(x),x),x)+x^2*diff(y(x),x)-3/4*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_1 \,x^{2} \left (1-\frac {1}{2} x +\frac {5}{32} x^{2}-\frac {7}{192} x^{3}+\frac {7}{1024} x^{4}-\frac {11}{10240} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (\ln \left (x \right ) \left (-\frac {1}{4} x^{2}+\frac {1}{8} x^{3}-\frac {5}{128} x^{4}+\frac {7}{768} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-2+x -\frac {1}{12} x^{3}+\frac {61}{1536} x^{4}-\frac {59}{5120} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )}{\sqrt {x}} \]

✓ Mathematica. Time used: 0.04 (sec). Leaf size: 103

ode=x^2*D[y[x],{x,2}]+x^2*D[y[x],x]-3/4*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (\frac {7 x^{11/2}}{1024}-\frac {7 x^{9/2}}{192}+\frac {5 x^{7/2}}{32}-\frac {x^{5/2}}{2}+x^{3/2}\right )+c_1 \left (\frac {1}{256} x^{3/2} \left (5 x^2-16 x+32\right ) \log (x)-\frac {91 x^4-224 x^3+192 x^2+1536 x-3072}{3072 \sqrt {x}}\right ) \]

✓ Sympy. Time used: 0.734 (sec). Leaf size: 31

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), x) + x**2*Derivative(y(x), (x, 2)) - 3*y(x)/4,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{1} x^{\frac {3}{2}} \left (- \frac {7 x^{3}}{192} + \frac {5 x^{2}}{32} - \frac {x}{2} + 1\right ) + O\left (x^{6}\right ) \]