Internal problem ID [6242]
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: 1.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{2} y^{\prime \prime }+3 x y^{\prime }+\left (x^{3}+x +1\right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.022 (sec). Leaf size: 81
Order:=8; dsolve(x^2*diff(y(x),x$2)+3*x*diff(y(x),x)+(1+x+x^3)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {\left (2 x -\frac {3}{4} x^{2}+\frac {19}{108} x^{3}-\frac {593}{3456} x^{4}+\frac {3629}{86400} x^{5}-\frac {7733}{1036800} x^{6}+\frac {485257}{118540800} x^{7}+\mathrm {O}\left (x^{8}\right )\right ) c_{2}+\left (\ln \relax (x ) c_{2}+c_{1}\right ) \left (1-x +\frac {1}{4} x^{2}-\frac {5}{36} x^{3}+\frac {41}{576} x^{4}-\frac {37}{2880} x^{5}+\frac {437}{103680} x^{6}-\frac {7817}{5080320} x^{7}+\mathrm {O}\left (x^{8}\right )\right )}{x} \]
✓ Solution by Mathematica
Time used: 0.055 (sec). Leaf size: 164
AsymptoticDSolveValue[x^2*y''[x]+3*x*y'[x]+(1+x+x^3)*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to \frac {c_1 \left (-\frac {7817 x^7}{5080320}+\frac {437 x^6}{103680}-\frac {37 x^5}{2880}+\frac {41 x^4}{576}-\frac {5 x^3}{36}+\frac {x^2}{4}-x+1\right )}{x}+c_2 \left (\frac {\frac {485257 x^7}{118540800}-\frac {7733 x^6}{1036800}+\frac {3629 x^5}{86400}-\frac {593 x^4}{3456}+\frac {19 x^3}{108}-\frac {3 x^2}{4}+2 x}{x}+\frac {\left (-\frac {7817 x^7}{5080320}+\frac {437 x^6}{103680}-\frac {37 x^5}{2880}+\frac {41 x^4}{576}-\frac {5 x^3}{36}+\frac {x^2}{4}-x+1\right ) \log (x)}{x}\right ) \]