43.2.17 problem 7.3.103

Internal problem ID [6874]
Book : Notes on Diffy Qs. Differential Equations for Engineers. By by Jiri Lebl, 2013.
Section : Chapter 7. POWER SERIES METHODS. 7.3.2 The method of Frobenius. Exercises. page 300
Problem number : 7.3.103
Date solved : Wednesday, March 05, 2025 at 02:47:29 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 62

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

\[ y = \frac {c_1 \,x^{2} \left (1-\frac {1}{3} x +\frac {1}{24} x^{2}-\frac {1}{360} x^{3}+\frac {1}{8640} x^{4}-\frac {1}{302400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (\ln \left (x \right ) \left (x^{2}-\frac {1}{3} x^{3}+\frac {1}{24} x^{4}-\frac {1}{360} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-2-2 x +\frac {4}{9} x^{3}-\frac {25}{288} x^{4}+\frac {157}{21600} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )}{\sqrt {x}} \]

✓ Mathematica. Time used: 0.027 (sec). Leaf size: 101

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

\[ y(x)\to c_2 \left (\frac {x^{11/2}}{8640}-\frac {x^{9/2}}{360}+\frac {x^{7/2}}{24}-\frac {x^{5/2}}{3}+x^{3/2}\right )+c_1 \left (\frac {31 x^4-176 x^3+144 x^2+576 x+576}{576 \sqrt {x}}-\frac {1}{48} x^{3/2} \left (x^2-8 x+24\right ) \log (x)\right ) \]

✓ Sympy. Time used: 0.953 (sec). Leaf size: 27

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

\[ y{\left (x \right )} = C_{1} x^{\frac {3}{2}} \left (- \frac {x^{3}}{360} + \frac {x^{2}}{24} - \frac {x}{3} + 1\right ) + O\left (x^{6}\right ) \]