50.18.11 problem 5

Internal problem ID [8105]
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 4.3. Second-Order Linear Equations: Ordinary Points. Page 169
Problem number : 5
Date solved : Monday, January 27, 2025 at 03:43:50 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 109

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

\[ y = \left (1-\frac {\left (2 p +1\right ) x^{2}}{4}+\frac {\left (4 p^{2}+4 p +3\right ) x^{4}}{96}-\frac {\left (8 p^{3}+12 p^{2}+34 p +15\right ) x^{6}}{5760}\right ) y \left (0\right )+\left (x -\frac {\left (2 p +1\right ) x^{3}}{12}+\frac {\left (4 p^{2}+4 p +7\right ) x^{5}}{480}-\frac {\left (8 p^{3}+12 p^{2}+58 p +27\right ) x^{7}}{40320}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \]

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 142

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

\[ y(x)\to c_2 \left (\frac {(-4 p-2) (4 p+2)^2 x^7}{322560}+\frac {13 (-4 p-2) x^7}{40320}+\frac {(4 p+2)^2 x^5}{1920}+\frac {1}{24} (-4 p-2) x^3+\frac {x^5}{80}+x\right )+c_1 \left (\frac {(-4 p-2) (4 p+2)^2 x^6}{46080}+\frac {7 (-4 p-2) x^6}{5760}+\frac {1}{384} (4 p+2)^2 x^4+\frac {1}{8} (-4 p-2) x^2+\frac {x^4}{48}+1\right ) \]