78.3.26 problem 7.d(1)

Internal problem ID [21022]
Book : A FIRST COURSE IN DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS. By Russell Herman. University of North Carolina Wilmington. LibreText. compiled on 06/09/2025
Section : Chapter 4, Series solutions. Problems section 4.9
Problem number : 7.d(1)
Date solved : Thursday, October 02, 2025 at 07:01:31 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.017 (sec). Leaf size: 45

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

\[ y = \frac {c_1 \left (1-x -\frac {1}{2} x^{2}-\frac {1}{18} x^{3}-\frac {1}{360} x^{4}-\frac {1}{12600} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{\sqrt {x}}+c_2 x \left (1+\frac {1}{5} x +\frac {1}{70} x^{2}+\frac {1}{1890} x^{3}+\frac {1}{83160} x^{4}+\frac {1}{5405400} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 84

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

\[ y(x)\to c_1 x \left (\frac {x^5}{5405400}+\frac {x^4}{83160}+\frac {x^3}{1890}+\frac {x^2}{70}+\frac {x}{5}+1\right )+\frac {c_2 \left (-\frac {x^5}{12600}-\frac {x^4}{360}-\frac {x^3}{18}-\frac {x^2}{2}-x+1\right )}{\sqrt {x}} \]

✓ Sympy. Time used: 0.417 (sec). Leaf size: 60

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

\[ y{\left (x \right )} = C_{2} x \left (\frac {x^{4}}{83160} + \frac {x^{3}}{1890} + \frac {x^{2}}{70} + \frac {x}{5} + 1\right ) + \frac {C_{1} \left (- \frac {x^{5}}{12600} - \frac {x^{4}}{360} - \frac {x^{3}}{18} - \frac {x^{2}}{2} - x + 1\right )}{\sqrt {x}} + O\left (x^{6}\right ) \]