49.17.1 problem 1(a)

Internal problem ID [7708]
Book : An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY 1961
Section : Chapter 4. Linear equations with Regular Singular Points. Page 154
Problem number : 1(a)
Date solved : Monday, January 27, 2025 at 03:10:18 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+\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: 0.013 (sec). Leaf size: 53

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

\[ y = c_{1} x \left (1-\frac {1}{3} x +\frac {1}{12} x^{2}-\frac {1}{60} x^{3}+\frac {1}{360} x^{4}-\frac {1}{2520} x^{5}+\frac {1}{20160} x^{6}-\frac {1}{181440} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\frac {c_{2} \left (-2+2 x -x^{2}+\frac {1}{3} x^{3}-\frac {1}{12} x^{4}+\frac {1}{60} x^{5}-\frac {1}{360} x^{6}+\frac {1}{2520} x^{7}+\operatorname {O}\left (x^{8}\right )\right )}{x} \]

✓ Solution by Mathematica

Time used: 0.041 (sec). Leaf size: 92

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

\[ y(x)\to c_1 \left (\frac {x^5}{720}-\frac {x^4}{120}+\frac {x^3}{24}-\frac {x^2}{6}+\frac {x}{2}+\frac {1}{x}-1\right )+c_2 \left (\frac {x^7}{20160}-\frac {x^6}{2520}+\frac {x^5}{360}-\frac {x^4}{60}+\frac {x^3}{12}-\frac {x^2}{3}+x\right ) \]