Internal problem ID [6246]
Book: Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition.
1997.
Section: CHAPTER 18. Power series solutions. 18.11 Many-Term Recurrence Relations. Exercises page
391
Problem number: 6.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x y^{\prime \prime }+x y^{\prime }+\left (x^{4}+1\right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.042 (sec). Leaf size: 70
Order:=8; dsolve(x*diff(y(x),x$2)+x*diff(y(x),x)+(1+x^4)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} x \left (1-x +\frac {1}{2} x^{2}-\frac {1}{6} x^{3}+\frac {1}{24} x^{4}-\frac {1}{24} x^{5}+\frac {31}{1008} x^{6}-\frac {47}{3528} x^{7}+\mathrm {O}\left (x^{8}\right )\right )+c_{2} \left (\ln \relax (x ) \left (-x +x^{2}-\frac {1}{2} x^{3}+\frac {1}{6} x^{4}-\frac {1}{24} x^{5}+\frac {1}{24} x^{6}-\frac {31}{1008} x^{7}+\mathrm {O}\left (x^{8}\right )\right )+\left (1-x +\frac {1}{4} x^{3}-\frac {5}{36} x^{4}-\frac {7}{1440} x^{5}+\frac {49}{2400} x^{6}+\frac {10847}{2116800} x^{7}+\mathrm {O}\left (x^{8}\right )\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.063 (sec). Leaf size: 114
AsymptoticDSolveValue[x*y''[x]+x*y'[x]+(1+x^4)*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 \left (\frac {1}{24} x \left (x^5-x^4+4 x^3-12 x^2+24 x-24\right ) \log (x)+\frac {-153 x^6+265 x^5-2200 x^4+5400 x^3-7200 x^2+7200}{7200}\right )+c_2 \left (\frac {31 x^7}{1008}-\frac {x^6}{24}+\frac {x^5}{24}-\frac {x^4}{6}+\frac {x^3}{2}-x^2+x\right ) \]