23.5.11 problem 3(j)

Internal problem ID [4188]
Book : Theory and solutions of Ordinary Differential equations, Donald Greenspan, 1960
Section : Chapter 7. Special functions. Exercises at page 124
Problem number : 3(j)
Date solved : Monday, January 27, 2025 at 08:41:33 AM
CAS classification : [_Jacobi]

\begin{align*} y^{\prime \prime }+\frac {\left (1-5 x \right ) y^{\prime }}{-x^{2}+x}-\frac {4 y}{-x^{2}+x}&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.014 (sec). Leaf size: 44

Order:=6; 
dsolve(diff(y(x),x$2)+(1-5*x)/(x-x^2)*diff(y(x),x)-4/(x-x^2)*y(x) =0,y(x),type='series',x=0);
 

\[ y \left (x \right ) = \left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1+4 x +9 x^{2}+16 x^{3}+25 x^{4}+36 x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (\left (-4\right ) x -12 x^{2}-24 x^{3}-40 x^{4}-60 x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \]

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 88

AsymptoticDSolveValue[D[y[x],{x,2}]+(1-5*x)/(x-x^2)*D[y[x],x]-4/(x-x^2)*y[x] ==0,{y[x]},{x,0,"6"-1}]
 

\[ \{y(x)\}\to c_1 \left (36 x^5+25 x^4+16 x^3+9 x^2+4 x+1\right )+c_2 \left (-60 x^5-40 x^4-24 x^3-12 x^2+\left (36 x^5+25 x^4+16 x^3+9 x^2+4 x+1\right ) \log (x)-4 x\right ) \]