68.17.13 problem 13

Internal problem ID [17831]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.9, page 215
Problem number : 13
Date solved : Thursday, October 02, 2025 at 02:28:40 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+\left (\frac {1}{2 x}-2\right ) y^{\prime }-\frac {35 y}{16 x^{2}}&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.016 (sec). Leaf size: 62

Order:=6; 
ode:=diff(diff(y(x),x),x)+(1/2/x-2)*diff(y(x),x)-35/16/x^2*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_1 \,x^{3} \left (1+\frac {7}{8} x +\frac {77}{160} x^{2}+\frac {77}{384} x^{3}+\frac {209}{3072} x^{4}+\frac {4807}{245760} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (\ln \left (x \right ) \left (\frac {15}{8} x^{3}+\frac {105}{64} x^{4}+\frac {231}{256} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (12+15 x +\frac {15}{4} x^{2}-\frac {13}{2} x^{3}-\frac {1741}{256} x^{4}-\frac {4141}{1024} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x^{{5}/{4}}} \]

✓ Mathematica. Time used: 0.028 (sec). Leaf size: 98

ode=D[y[x],{x,2}]+(1/2*1/x-2)*D[y[x],x]-35/16*1/x^2*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (\frac {209 x^{23/4}}{3072}+\frac {77 x^{19/4}}{384}+\frac {77 x^{15/4}}{160}+\frac {7 x^{11/4}}{8}+x^{7/4}\right )+c_1 \left (\frac {5}{256} x^{7/4} (7 x+8) \log (x)-\frac {627 x^4+608 x^3-320 x^2-1280 x-1024}{1024 x^{5/4}}\right ) \]

✓ Sympy. Time used: 0.418 (sec). Leaf size: 32

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((-2 + 1/(2*x))*Derivative(y(x), x) + Derivative(y(x), (x, 2)) - 35*y(x)/(16*x**2),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{1} x^{\frac {7}{4}} \left (\frac {77 x^{3}}{384} + \frac {77 x^{2}}{160} + \frac {7 x}{8} + 1\right ) + O\left (x^{6}\right ) \]