85.77.5 problem 8 (b)

Internal problem ID [22988]
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 341
Problem number : 8 (b)
Date solved : Thursday, October 02, 2025 at 09:17:17 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x^{2} y^{\prime \prime }+4 x y^{\prime }+\left (2 x^{2}-1\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.012 (sec). Leaf size: 34

Order:=6; 
ode:=4*x^2*diff(diff(y(x),x),x)+4*x*diff(y(x),x)+(2*x^2-1)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_1 x \left (1-\frac {1}{12} x^{2}+\frac {1}{480} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (1-\frac {1}{4} x^{2}+\frac {1}{96} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{\sqrt {x}} \]

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 58

ode=4*x^2*D[y[x],{x,2}]+4*x*D[y[x],x]+(2*x^2-1)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (\frac {x^{7/2}}{96}-\frac {x^{3/2}}{4}+\frac {1}{\sqrt {x}}\right )+c_2 \left (\frac {x^{9/2}}{480}-\frac {x^{5/2}}{12}+\sqrt {x}\right ) \]

✓ Sympy. Time used: 0.377 (sec). Leaf size: 42

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*x**2*Derivative(y(x), (x, 2)) + 4*x*Derivative(y(x), x) + (2*x**2 - 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \sqrt {x} \left (\frac {x^{4}}{480} - \frac {x^{2}}{12} + 1\right ) + \frac {C_{1} \left (\frac {x^{4}}{96} - \frac {x^{2}}{4} + 1\right )}{\sqrt {x}} + O\left (x^{6}\right ) \]