Internal problem ID [5838]
Book: DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL,
WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th
edition.
Section: CHAPTER 6 SERIES SOLUTIONS OF LINEAR EQUATIONS. 6.3 SOLUTIONS ABOUT
SINGULAR POINTS. EXERCISES 6.3. Page 255
Problem number: 8.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x \left (x^{2}+1\right )^{2} y^{\prime \prime }+y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.094 (sec). Leaf size: 70
Order:=8; dsolve(x*(x^2+1)^2*diff(y(x),x$2)+y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} x \left (1-\frac {1}{2} x +\frac {1}{12} x^{2}+\frac {23}{144} x^{3}-\frac {167}{2880} x^{4}-\frac {7993}{86400} x^{5}+\frac {23599}{518400} x^{6}+\frac {1860281}{29030400} x^{7}+\mathrm {O}\left (x^{8}\right )\right )+c_{2} \left (\ln \relax (x ) \left (-x +\frac {1}{2} x^{2}-\frac {1}{12} x^{3}-\frac {23}{144} x^{4}+\frac {167}{2880} x^{5}+\frac {7993}{86400} x^{6}-\frac {23599}{518400} x^{7}+\mathrm {O}\left (x^{8}\right )\right )+\left (1-\frac {3}{4} x^{2}+\frac {19}{36} x^{3}+\frac {85}{1728} x^{4}-\frac {21907}{86400} x^{5}+\frac {787}{81000} x^{6}+\frac {5987917}{36288000} x^{7}+\mathrm {O}\left (x^{8}\right )\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.13 (sec). Leaf size: 121
AsymptoticDSolveValue[x*(x^2+1)^2*y''[x]+y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 \left (\frac {x \left (7993 x^5+5010 x^4-13800 x^3-7200 x^2+43200 x-86400\right ) \log (x)}{86400}+\frac {-107303 x^6-403755 x^5+270750 x^4+792000 x^3-1620000 x^2+1296000 x+1296000}{1296000}\right )+c_2 \left (\frac {23599 x^7}{518400}-\frac {7993 x^6}{86400}-\frac {167 x^5}{2880}+\frac {23 x^4}{144}+\frac {x^3}{12}-\frac {x^2}{2}+x\right ) \]