54.5.1 problem 1

Internal problem ID [8616]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 18. Power series solutions. 18.6. Indicial Equation with Equal Roots. Exercises page 373
Problem number : 1
Date solved : Monday, January 27, 2025 at 04:17:32 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-x \left (1+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: 54

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

\[ y = x \left (\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1+x +\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{120} x^{5}+\frac {1}{720} x^{6}+\frac {1}{5040} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (-x -\frac {3}{4} x^{2}-\frac {11}{36} x^{3}-\frac {25}{288} x^{4}-\frac {137}{7200} x^{5}-\frac {49}{14400} x^{6}-\frac {121}{235200} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_{2} \right ) \]

✓ Solution by Mathematica

Time used: 0.010 (sec). Leaf size: 154

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

\[ y(x)\to c_1 x \left (\frac {x^7}{5040}+\frac {x^6}{720}+\frac {x^5}{120}+\frac {x^4}{24}+\frac {x^3}{6}+\frac {x^2}{2}+x+1\right )+c_2 \left (x \left (-\frac {121 x^7}{235200}-\frac {49 x^6}{14400}-\frac {137 x^5}{7200}-\frac {25 x^4}{288}-\frac {11 x^3}{36}-\frac {3 x^2}{4}-x\right )+x \left (\frac {x^7}{5040}+\frac {x^6}{720}+\frac {x^5}{120}+\frac {x^4}{24}+\frac {x^3}{6}+\frac {x^2}{2}+x+1\right ) \log (x)\right ) \]