85.77.4 problem 8 (a)

Internal problem ID [22987]
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 (a)
Date solved : Thursday, October 02, 2025 at 09:17:16 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+x y^{\prime }+\left (3 x^{2}-4\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.013 (sec). Leaf size: 44

Order:=6; 
ode:=x^2*diff(diff(y(x),x),x)+x*diff(y(x),x)+(3*x^2-4)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_1 \,x^{4} \left (1-\frac {1}{4} x^{2}+\frac {3}{128} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (\ln \left (x \right ) \left (81 x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (-144-108 x^{2}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x^{2}} \]

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 54

ode=x^2*D[y[x],{x,2}]+x*D[y[x],x]+(3*x^2-4)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (\frac {\left (3 x^2+8\right )^2}{64 x^2}-\frac {9}{16} x^2 \log (x)\right )+c_2 \left (\frac {3 x^6}{128}-\frac {x^4}{4}+x^2\right ) \]

✓ Sympy. Time used: 0.325 (sec). Leaf size: 17

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + x*Derivative(y(x), x) + (3*x**2 - 4)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

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