64.15.20 problem 20

Internal problem ID [13597]
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 : 20
Date solved : Tuesday, January 28, 2025 at 05:53:37 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.061 (sec). Leaf size: 44

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

\[ y = \frac {c_{1} x^{4} \left (1-\frac {1}{4} x^{2}+\frac {5}{128} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (\left (-9\right ) x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (-144+36 x^{2}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x} \]

✓ Solution by Mathematica

Time used: 0.016 (sec). Leaf size: 55

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

\[ y(x)\to c_2 \left (\frac {5 x^7}{128}-\frac {x^5}{4}+x^3\right )+c_1 \left (\frac {1}{16} x^3 \log (x)-\frac {x^4+16 x^2-64}{64 x}\right ) \]