85.75.2 problem 1 (b)

Internal problem ID [22967]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter 7. Solution of differential equations by use of series. A Exercises at page 329
Problem number : 1 (b)
Date solved : Thursday, October 02, 2025 at 09:17:02 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.016 (sec). Leaf size: 32

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

\[ y = c_1 \sqrt {x}\, \left (1+\frac {1}{10} x^{2}+\frac {1}{360} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (1+\frac {1}{6} x^{2}+\frac {1}{168} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 47

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

\[ y(x)\to c_1 \sqrt {x} \left (\frac {x^4}{360}+\frac {x^2}{10}+1\right )+c_2 \left (\frac {x^4}{168}+\frac {x^2}{6}+1\right ) \]

✓ Sympy. Time used: 0.303 (sec). Leaf size: 37

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

\[ y{\left (x \right )} = C_{2} \left (\frac {x^{4}}{168} + \frac {x^{2}}{6} + 1\right ) + C_{1} \sqrt {x} \left (\frac {x^{4}}{360} + \frac {x^{2}}{10} + 1\right ) + O\left (x^{6}\right ) \]