49.18.1 problem 1(a)

Internal problem ID [7718]
Book : An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY 1961
Section : Chapter 4. Linear equations with Regular Singular Points. Page 159
Problem number : 1(a)
Date solved : Monday, January 27, 2025 at 03:10:32 PM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} 3 x^{2} y^{\prime \prime }+5 x y^{\prime }+3 y x&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.012 (sec). Leaf size: 52

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

\[ y = \frac {c_{1} \left (1-3 x +\frac {9}{8} x^{2}-\frac {9}{56} x^{3}+\frac {27}{2240} x^{4}-\frac {81}{145600} x^{5}+\frac {81}{4659200} x^{6}-\frac {243}{619673600} x^{7}+\operatorname {O}\left (x^{8}\right )\right )}{x^{{2}/{3}}}+c_{2} \left (1-\frac {3}{5} x +\frac {9}{80} x^{2}-\frac {9}{880} x^{3}+\frac {27}{49280} x^{4}-\frac {81}{4188800} x^{5}+\frac {81}{167552000} x^{6}-\frac {243}{26975872000} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 111

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

\[ y(x)\to c_1 \left (-\frac {243 x^7}{26975872000}+\frac {81 x^6}{167552000}-\frac {81 x^5}{4188800}+\frac {27 x^4}{49280}-\frac {9 x^3}{880}+\frac {9 x^2}{80}-\frac {3 x}{5}+1\right )+\frac {c_2 \left (-\frac {243 x^7}{619673600}+\frac {81 x^6}{4659200}-\frac {81 x^5}{145600}+\frac {27 x^4}{2240}-\frac {9 x^3}{56}+\frac {9 x^2}{8}-3 x+1\right )}{x^{2/3}} \]