41.1.7 problem Ex. 6(vi), page 257

Internal problem ID [6820]
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 : Monday, January 27, 2025 at 02:31:08 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*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 28

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

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

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 40

AsymptoticDSolveValue[x*D[y[x],{x,2}]+(4*x^2+1)*D[y[x],x]+4*x*(x^2+1)*y[x]==0,y[x],{x,0,"6"-1}]
 

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