7.14.4 problem 4

Internal problem ID [429]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 3. Power series methods. Section 3.2 (Series solution near ordinary points). Problems at page 216
Problem number : 4
Date solved : Monday, January 27, 2025 at 02:53:35 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} \left (x^{2}+1\right ) y^{\prime \prime }+6 x y^{\prime }+4 y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 39

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

\[ y = \left (3 x^{4}-2 x^{2}+1\right ) y \left (0\right )+\left (x -\frac {5}{3} x^{3}+\frac {7}{3} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 38

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

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