49.19.6 problem 3(c)

Internal problem ID [7726]
Book : An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY 1961
Section : Chapter 4. Linear equations with Regular Singular Points. Page 166
Problem number : 3(c)
Date solved : Monday, January 27, 2025 at 03:10:44 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+5 x y^{\prime }+\left (-x^{3}+3\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.016 (sec). Leaf size: 35

Order:=8; 
dsolve(x^2*diff(y(x),x$2)+5*x*diff(y(x),x)+(3-x^3)*y(x)=0,y(x),type='series',x=0);
 

\[ y = \frac {c_{1} \left (1+\frac {1}{15} x^{3}+\frac {1}{720} x^{6}+\operatorname {O}\left (x^{8}\right )\right )}{x}+\frac {c_{2} \left (-2-\frac {2}{3} x^{3}-\frac {1}{36} x^{6}+\operatorname {O}\left (x^{8}\right )\right )}{x^{3}} \]

✓ Solution by Mathematica

Time used: 0.013 (sec). Leaf size: 40

AsymptoticDSolveValue[x^2*D[y[x],{x,2}]+5*x*D[y[x],x]+(3-3*x^3)*y[x]==0,y[x],{x,0,"8"-1}]
 

\[ y(x)\to c_1 \left (\frac {x^3}{8}+\frac {1}{x^3}+1\right )+c_2 \left (\frac {x^5}{80}+\frac {x^2}{5}+\frac {1}{x}\right ) \]