7.16.15 problem 16

Internal problem ID [512]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 3. Power series methods. Section 3.4 (Method of Frobenius: The exceptional cases). Problems at page 246
Problem number : 16
Date solved : Monday, January 27, 2025 at 02:54:17 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {9}{4}\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: 36

Order:=6; 
dsolve(x^2*diff(y(x),x$2)+x*diff(y(x),x)+(x^2-9/4)*y(x)=0,y(x),type='series',x=0);
 

\[ y = \frac {c_1 \,x^{3} \left (1-\frac {1}{10} x^{2}+\frac {1}{280} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (12+6 x^{2}-\frac {3}{2} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{{3}/{2}}} \]

✓ Solution by Mathematica

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

AsymptoticDSolveValue[x^2*D[y[x],{x,2}]+x*D[y[x],x]+(x^2-9/4)*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_1 \left (-\frac {x^{5/2}}{8}+\frac {1}{x^{3/2}}+\frac {\sqrt {x}}{2}\right )+c_2 \left (\frac {x^{11/2}}{280}-\frac {x^{7/2}}{10}+x^{3/2}\right ) \]