49.19.7 problem 3(d)

Internal problem ID [7727]
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(d)
Date solved : Monday, January 27, 2025 at 03:10:45 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-2 x \left (1+x \right ) y^{\prime }+2 y \left (1+x \right )&=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)-2*x*(x+1)*diff(y(x),x)+2*(x+1)*y(x)=0,y(x),type='series',x=0);
 

\[ y = c_{1} x^{2} \left (1+x +\frac {2}{3} x^{2}+\frac {1}{3} x^{3}+\frac {2}{15} x^{4}+\frac {2}{45} x^{5}+\frac {4}{315} x^{6}+\frac {1}{315} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} x \left (1+2 x +2 x^{2}+\frac {4}{3} x^{3}+\frac {2}{3} x^{4}+\frac {4}{15} x^{5}+\frac {4}{45} x^{6}+\frac {8}{315} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.089 (sec). Leaf size: 92

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

\[ y(x)\to c_1 \left (\frac {4 x^7}{45}+\frac {4 x^6}{15}+\frac {2 x^5}{3}+\frac {4 x^4}{3}+2 x^3+2 x^2+x\right )+c_2 \left (\frac {4 x^8}{315}+\frac {2 x^7}{45}+\frac {2 x^6}{15}+\frac {x^5}{3}+\frac {2 x^4}{3}+x^3+x^2\right ) \]