7.15.39 problem 40

Internal problem ID [495]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 3. Power series methods. Section 3.3 (Regular singular points). Problems at page 231
Problem number : 40
Date solved : Wednesday, February 05, 2025 at 03:42:14 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 47

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

\[ y = c_1 \,x^{-i} \left (1+\left (\frac {1}{5}+\frac {2 i}{5}\right ) x +\left (-\frac {1}{40}+\frac {3 i}{40}\right ) x^{2}+\left (-\frac {3}{520}+\frac {7 i}{1560}\right ) x^{3}+\left (-\frac {1}{2496}+\frac {i}{12480}\right ) x^{4}+\left (-\frac {9}{603200}-\frac {i}{361920}\right ) x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \,x^{i} \left (1+\left (\frac {1}{5}-\frac {2 i}{5}\right ) x +\left (-\frac {1}{40}-\frac {3 i}{40}\right ) x^{2}+\left (-\frac {3}{520}-\frac {7 i}{1560}\right ) x^{3}+\left (-\frac {1}{2496}-\frac {i}{12480}\right ) x^{4}+\left (-\frac {9}{603200}+\frac {i}{361920}\right ) x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.035 (sec). Leaf size: 90

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

\[ y(x)\to \left (\frac {1}{12480}+\frac {i}{2496}\right ) c_2 x^{-i} \left (i x^4+(8+16 i) x^3+(168+96 i) x^2+(1056-288 i) x+(480-2400 i)\right )-\left (\frac {1}{2496}+\frac {i}{12480}\right ) c_1 x^i \left (x^4+(16+8 i) x^3+(96+168 i) x^2-(288-1056 i) x-(2400-480 i)\right ) \]