35.1.7 problem Ex. 6(vi), page 257

Internal problem ID [8106]
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. 6(vi), page 257
Date solved : Tuesday, September 30, 2025 at 05:15:37 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.023 (sec). Leaf size: 28

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

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

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 40

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

\[ y(x)\to c_1 \left (\frac {x^4}{2}-x^2+1\right )+c_2 \left (\frac {x^4}{2}-x^2+1\right ) \log (x) \]

✓ Sympy. Time used: 0.280 (sec). Leaf size: 17

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

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