7.15.31 problem 31

Internal problem ID [487]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 3. Power series methods. Section 3.3 (Regular singular points). Problems at page 231
Problem number : 31
Date solved : Monday, January 27, 2025 at 02:54:03 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x^{2} y^{\prime \prime }-4 x y^{\prime }+\left (-4 x^{2}+3\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 34

Order:=6; 
dsolve(4*x^2*diff(y(x),x$2)-4*x*diff(y(x),x)+(3-4*x^2)*y(x)=0,y(x),type='series',x=0);
 

\[ y = \sqrt {x}\, \left (\left (1+\frac {1}{6} x^{2}+\frac {1}{120} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) x c_1 +\left (1+\frac {1}{2} x^{2}+\frac {1}{24} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \right ) \]

✓ Solution by Mathematica

Time used: 0.010 (sec). Leaf size: 58

AsymptoticDSolveValue[4*x^2*D[y[x],{x,2}]-4*x*D[y[x],x]+(3-4*x^2)*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_1 \left (\frac {x^{9/2}}{24}+\frac {x^{5/2}}{2}+\sqrt {x}\right )+c_2 \left (\frac {x^{11/2}}{120}+\frac {x^{7/2}}{6}+x^{3/2}\right ) \]