54.7.7 problem 7

Internal problem ID [8655]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 18. Power series solutions. 18.9 Indicial Equation with Difference of Roots a Positive Integer: Logarithmic Case. Exercises page 384
Problem number : 7
Date solved : Monday, January 27, 2025 at 04:18:28 PM
CAS classification : [_Jacobi]

\begin{align*} x \left (1-x \right ) y^{\prime \prime }+2 \left (1-x \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: 52

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

\[ y = \frac {\ln \left (x \right ) \left (\left (-2\right ) x +2 x^{2}+\operatorname {O}\left (x^{8}\right )\right ) c_{2} +c_{1} \left (1-x +\operatorname {O}\left (x^{8}\right )\right ) x +\left (1-4 x^{2}+x^{3}+\frac {1}{3} x^{4}+\frac {1}{6} x^{5}+\frac {1}{10} x^{6}+\frac {1}{15} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_{2}}{x} \]

✓ Solution by Mathematica

Time used: 0.382 (sec). Leaf size: 60

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

\[ y(x)\to c_1 \left (\frac {3 x^6+5 x^5+10 x^4+30 x^3-150 x^2+30 x+30}{30 x}+2 (x-1) \log (x)\right )+c_2 (1-x) \]