50.18.14 problem 8

Internal problem ID [8108]
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 : 8
Date solved : Monday, January 27, 2025 at 03:43:54 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-2 x y^{\prime }+2 p 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: 85

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

\[ y = \left (1-p \,x^{2}+\frac {p \left (p -2\right ) x^{4}}{6}-\frac {p \left (p -2\right ) \left (p -4\right ) x^{6}}{90}\right ) y \left (0\right )+\left (x -\frac {\left (p -1\right ) x^{3}}{3}+\frac {\left (p^{2}-4 p +3\right ) x^{5}}{30}-\frac {\left (p^{3}-9 p^{2}+23 p -15\right ) x^{7}}{630}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \]

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 141

AsymptoticDSolveValue[D[y[x],{x,2}]-2*x*D[y[x],x]+2*p*y[x]==0,y[x],{x,0,"8"-1}]
 

\[ y(x)\to c_2 \left (-\frac {1}{630} p^3 x^7+\frac {p^2 x^7}{70}+\frac {p^2 x^5}{30}-\frac {23 p x^7}{630}-\frac {2 p x^5}{15}-\frac {p x^3}{3}+\frac {x^7}{42}+\frac {x^5}{10}+\frac {x^3}{3}+x\right )+c_1 \left (-\frac {1}{90} p^3 x^6+\frac {p^2 x^6}{15}+\frac {p^2 x^4}{6}-\frac {4 p x^6}{45}-\frac {p x^4}{3}-p x^2+1\right ) \]