64.15.17 problem 17

Internal problem ID [13594]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 6, Series solutions of linear differential equations. Section 6.2 (Frobenius). Exercises page 251
Problem number : 17
Date solved : Tuesday, January 28, 2025 at 05:53:33 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (2 x^{2}-x \right ) y^{\prime \prime }+\left (2 x -2\right ) y^{\prime }+\left (-2 x^{2}+3 x -2\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.057 (sec). Leaf size: 44

Order:=6; 
dsolve((2*x^2-x)*diff(y(x),x$2)+(2*x-2)*diff(y(x),x)+(-2*x^2+3*x-2)*y(x)=0,y(x),type='series',x=0);
 

\[ y = c_{1} \left (1-x +\frac {1}{2} x^{2}-\frac {1}{6} x^{3}+\frac {1}{24} x^{4}-\frac {1}{120} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (1-2 x +\frac {7}{2} x^{2}-\frac {4}{3} x^{3}+\frac {13}{24} x^{4}-\frac {7}{60} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

✓ Solution by Mathematica

Time used: 0.047 (sec). Leaf size: 60

AsymptoticDSolveValue[(2*x^2-x)*D[y[x],{x,2}]+(2*x-2)*D[y[x],x]+(-2*x^2+3*x-2)*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_1 \left (\frac {7 x^3}{8}-\frac {7 x^2}{3}+\frac {11 x}{2}+\frac {1}{x}-4\right )+c_2 \left (\frac {x^4}{24}-\frac {x^3}{6}+\frac {x^2}{2}-x+1\right ) \]