50.20.2 problem 2

Internal problem ID [8130]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 4. Power Series Solutions and Special Functions. Section 4.5. More on Regular Singular Points. Page 183
Problem number : 2
Date solved : Monday, January 27, 2025 at 03:44:23 PM
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: 0.016 (sec). Leaf size: 52

Order:=8; 
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);
 

\[ y = \sqrt {x}\, \left (1+x +\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{120} x^{5}+\frac {1}{720} x^{6}+\frac {1}{5040} x^{7}\right ) \left (c_{2} \ln \left (x \right )+c_{1} \right )+O\left (x^{8}\right ) \]

✓ Solution by Mathematica

Time used: 0.010 (sec). Leaf size: 112

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,"8"-1}]
 

\[ y(x)\to c_1 \sqrt {x} \left (\frac {x^7}{5040}+\frac {x^6}{720}+\frac {x^5}{120}+\frac {x^4}{24}+\frac {x^3}{6}+\frac {x^2}{2}+x+1\right )+c_2 \sqrt {x} \left (\frac {x^7}{5040}+\frac {x^6}{720}+\frac {x^5}{120}+\frac {x^4}{24}+\frac {x^3}{6}+\frac {x^2}{2}+x+1\right ) \log (x) \]