50.20.3 problem 3(a)

Internal problem ID [8131]
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 : 3(a)
Date solved : Monday, January 27, 2025 at 03:44:24 PM
CAS classification : [_Lienard]

\begin{align*} x y^{\prime \prime }+2 y^{\prime }+y x&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.015 (sec). Leaf size: 36

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

\[ y = c_{1} \left (1-\frac {1}{6} x^{2}+\frac {1}{120} x^{4}-\frac {1}{5040} x^{6}+\operatorname {O}\left (x^{8}\right )\right )+\frac {c_{2} \left (1-\frac {1}{2} x^{2}+\frac {1}{24} x^{4}-\frac {1}{720} x^{6}+\operatorname {O}\left (x^{8}\right )\right )}{x} \]

✓ Solution by Mathematica

Time used: 0.015 (sec). Leaf size: 56

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

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