78.20.2 problem 2

Internal problem ID [18439]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 5. Power Series Solutions and Special Functions. Section 30. Regular singular Points (continued). Problems at page 235
Problem number : 2
Date solved : Tuesday, January 28, 2025 at 11:50:03 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x^{2} y^{\prime \prime }-8 x^{2} y^{\prime }+\left (4 x^{2}+1\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 42

Order:=6; 
dsolve(4*x^2*diff(y(x),x$2)-8*x^2*diff(y(x),x)+(4*x^2+1)*y(x)=0,y(x),type='series',x=0);
 

\[ y = \sqrt {x}\, \left (1+x +\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{120} x^{5}\right ) \left (c_{2} \ln \left (x \right )+c_{1} \right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 84

AsymptoticDSolveValue[4*x^2*D[y[x],{x,2}]-8*x^2*D[y[x],x]+(4*x^2+1)*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_1 \sqrt {x} \left (\frac {x^5}{120}+\frac {x^4}{24}+\frac {x^3}{6}+\frac {x^2}{2}+x+1\right )+c_2 \sqrt {x} \left (\frac {x^5}{120}+\frac {x^4}{24}+\frac {x^3}{6}+\frac {x^2}{2}+x+1\right ) \log (x) \]