45.2.8 problem 8

Internal problem ID [7231]
Book : A FIRST COURSE IN DIFFERENTIAL EQUATIONS with Modeling Applications. Dennis G. Zill. 9th edition. Brooks/Cole. CA, USA.
Section : Chapter 6. SERIES SOLUTIONS OF LINEAR EQUATIONS. Exercises. 6.2 page 239
Problem number : 8
Date solved : Monday, January 27, 2025 at 02:48:36 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.010 (sec). Leaf size: 58

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

\[ y = c_{1} x \left (1-\frac {1}{2} x +\frac {1}{12} x^{2}+\frac {23}{144} x^{3}-\frac {167}{2880} x^{4}-\frac {7993}{86400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (-x +\frac {1}{2} x^{2}-\frac {1}{12} x^{3}-\frac {23}{144} x^{4}+\frac {167}{2880} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (1-\frac {3}{4} x^{2}+\frac {19}{36} x^{3}+\frac {85}{1728} x^{4}-\frac {21907}{86400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.026 (sec). Leaf size: 87

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

\[ y(x)\to c_1 \left (\frac {361 x^4+1056 x^3-2160 x^2+1728 x+1728}{1728}-\frac {1}{144} x \left (23 x^3+12 x^2-72 x+144\right ) \log (x)\right )+c_2 \left (-\frac {167 x^5}{2880}+\frac {23 x^4}{144}+\frac {x^3}{12}-\frac {x^2}{2}+x\right ) \]