64.15.25 problem 25

Internal problem ID [13602]
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 : 25
Date solved : Tuesday, January 28, 2025 at 05:53:44 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.038 (sec). Leaf size: 34

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

\[ y = \left (\left (\ln \left (x \right ) c_{2} +c_{1} \right ) \left (1-\frac {1}{4} x^{2}+\frac {1}{64} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (\frac {1}{4} x^{2}-\frac {3}{128} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \right ) x \]

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 65

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

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