85.78.1 problem 1

Internal problem ID [22989]
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. B Exercises at page 341
Problem number : 1
Date solved : Thursday, October 02, 2025 at 09:17:18 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.009 (sec). Leaf size: 46

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

\[ y = \frac {c_1 \,x^{2} \left (1-\frac {1}{2} x^{2}+\frac {1}{12} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (\ln \left (x \right ) \left (4 x^{2}-2 x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (-2+\frac {3}{2} x^{4}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x} \]

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 55

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

\[ y(x)\to c_2 \left (\frac {x^5}{12}-\frac {x^3}{2}+x\right )+c_1 \left (x \left (x^2-2\right ) \log (x)-\frac {5 x^4-4 x^2-4}{4 x}\right ) \]

✓ Sympy. Time used: 0.296 (sec). Leaf size: 20

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

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