54.5.8 problem 8

Internal problem ID [8623]
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 : 8
Date solved : Monday, January 27, 2025 at 04:17:41 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.021 (sec). Leaf size: 40

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

\[ y = \left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-x +\operatorname {O}\left (x^{8}\right )\right )+\left (\frac {5}{2} x -\frac {3}{8} x^{2}-\frac {1}{12} x^{3}-\frac {5}{192} x^{4}-\frac {3}{320} x^{5}-\frac {7}{1920} x^{6}-\frac {1}{672} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_{2} \]

✓ Solution by Mathematica

Time used: 0.023 (sec). Leaf size: 71

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

\[ y(x)\to c_2 \left (-\frac {x^7}{672}-\frac {7 x^6}{1920}-\frac {3 x^5}{320}-\frac {5 x^4}{192}-\frac {x^3}{12}-\frac {3 x^2}{8}+\frac {5 x}{2}+(1-x) \log (x)\right )+c_1 (1-x) \]