35.1.8 problem Ex. 8(i), page 258

Internal problem ID [8107]
Book : A treatise on Differential Equations by A. R. Forsyth. 6th edition. 1929. Macmillan Co. ltd. New York, reprinted 1956
Section : Chapter VI. Note I. Integration of linear equations in series by the method of Frobenius. page 243
Problem number : Ex. 8(i), page 258
Date solved : Tuesday, September 30, 2025 at 05:15:38 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.024 (sec). Leaf size: 407

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

\[ y = \sqrt {x}\, \left (c_1 \,x^{-\frac {\sqrt {1-16 a}}{2}} \left (1+4 \frac {1}{-1+\sqrt {1-16 a}} x +8 \frac {1}{\left (-1+\sqrt {1-16 a}\right ) \left (-2+\sqrt {1-16 a}\right )} x^{2}+\frac {32}{3} \frac {1}{\left (-1+\sqrt {1-16 a}\right ) \left (-2+\sqrt {1-16 a}\right ) \left (-3+\sqrt {1-16 a}\right )} x^{3}+\frac {32}{3} \frac {1}{\left (-1+\sqrt {1-16 a}\right ) \left (-2+\sqrt {1-16 a}\right ) \left (-3+\sqrt {1-16 a}\right ) \left (-4+\sqrt {1-16 a}\right )} x^{4}+\frac {128}{15} \frac {1}{\left (-1+\sqrt {1-16 a}\right ) \left (-2+\sqrt {1-16 a}\right ) \left (-3+\sqrt {1-16 a}\right ) \left (-4+\sqrt {1-16 a}\right ) \left (-5+\sqrt {1-16 a}\right )} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \,x^{\frac {\sqrt {1-16 a}}{2}} \left (1-4 \frac {1}{1+\sqrt {1-16 a}} x +8 \frac {1}{\left (1+\sqrt {1-16 a}\right ) \left (2+\sqrt {1-16 a}\right )} x^{2}-\frac {32}{3} \frac {1}{\left (1+\sqrt {1-16 a}\right ) \left (2+\sqrt {1-16 a}\right ) \left (3+\sqrt {1-16 a}\right )} x^{3}+\frac {32}{3} \frac {1}{\left (1+\sqrt {1-16 a}\right ) \left (2+\sqrt {1-16 a}\right ) \left (3+\sqrt {1-16 a}\right ) \left (4+\sqrt {1-16 a}\right )} x^{4}-\frac {128}{15} \frac {1}{\left (1+\sqrt {1-16 a}\right ) \left (2+\sqrt {1-16 a}\right ) \left (3+\sqrt {1-16 a}\right ) \left (4+\sqrt {1-16 a}\right ) \left (5+\sqrt {1-16 a}\right )} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right ) \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 1356

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

\begin{align*} \text {Solution too large to show}\end{align*}

✗ Sympy

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

ValueError : Expected Expr or iterable but got None