49.19.3 problem 1(iii)

Internal problem ID [7723]
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 : 1(iii)
Date solved : Monday, January 27, 2025 at 03:10:40 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (-x^{2}+1\right ) y^{\prime \prime }+3 \left (x^{2}+x \right ) y^{\prime }+y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.019 (sec). Leaf size: 56

Order:=8; 
dsolve((1-x^2)*x^2*diff(y(x),x$2)+3*(x+x^2)*diff(y(x),x)+y(x)=0,y(x),type='series',x=0);
 

\[ y = \frac {\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1+3 x +\frac {1}{2} x^{2}-\frac {1}{6} x^{3}+\frac {1}{16} x^{4}-\frac {43}{1200} x^{5}+\frac {161}{7200} x^{6}-\frac {1837}{117600} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (\left (-9\right ) x -\frac {7}{2} x^{2}+\frac {7}{9} x^{3}-\frac {25}{96} x^{4}+\frac {5141}{36000} x^{5}-\frac {2083}{24000} x^{6}+\frac {489941}{8232000} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_{2}}{x} \]

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 84

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

\[ y(x)\to c_2 \left (\frac {53 x^7}{630}+\frac {5 x^6}{24}+\frac {2 x^5}{15}-\frac {x^4}{4}-\frac {2 x^3}{3}+x\right )+c_1 \left (-\frac {19 x^7}{420}-\frac {x^6}{144}+\frac {3 x^5}{20}+\frac {5 x^4}{24}-\frac {x^2}{2}+1\right ) \]