7.16.16 problem 17

Internal problem ID [513]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 3. Power series methods. Section 3.4 (Method of Frobenius: The exceptional cases). Problems at page 246
Problem number : 17
Date solved : Monday, January 27, 2025 at 02:54:17 AM
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: 0.005 (sec). Leaf size: 46

Order:=6; 
dsolve(x^2*diff(y(x),x$2)-x*(1+x)*diff(y(x),x)+y(x)=0,y(x),type='series',x=0);
 

\[ y = x \left (\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1+x +\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{120} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-x -\frac {3}{4} x^{2}-\frac {11}{36} x^{3}-\frac {25}{288} x^{4}-\frac {137}{7200} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \right ) \]

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 112

AsymptoticDSolveValue[x^2*D[y[x],{x,2}]-x*(1+x)*D[y[x],x]+y[x]==0,y[x],{x,0,"6"-1}]
 

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