36.5.17 problem 18

Internal problem ID [6361]
Book : Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson 2018.
Section : Chapter 8, Series solutions of differential equations. Section 8.3. page 443
Problem number : 18
Date solved : Wednesday, March 05, 2025 at 12:36:47 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 38

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

\[ y = \left (1+\frac {1}{6} x^{2}+\frac {1}{27} x^{3}+\frac {5}{648} x^{4}+\frac {1}{540} x^{5}\right ) y \left (0\right )+y^{\prime }\left (0\right ) x +O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 41

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

\[ y(x)\to c_1 \left (\frac {x^5}{540}+\frac {5 x^4}{648}+\frac {x^3}{27}+\frac {x^2}{6}+1\right )+c_2 x \]

✓ Sympy. Time used: 0.762 (sec). Leaf size: 29

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

\[ y{\left (x \right )} = C_{2} \left (\frac {5 x^{4}}{648} + \frac {x^{3}}{27} + \frac {x^{2}}{6} + 1\right ) + C_{1} x + O\left (x^{6}\right ) \]