Internal problem ID [6244]
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: 3.
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 }+y^{\prime }+x \left (1+x \right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.062 (sec). Leaf size: 65
Order:=8; dsolve(x*diff(y(x),x$2)+diff(y(x),x)+x*(1+x)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \left (\frac {1}{4} x^{2}+\frac {2}{27} x^{3}-\frac {3}{128} x^{4}-\frac {253}{13500} x^{5}-\frac {95}{41472} x^{6}+\frac {153527}{148176000} x^{7}+\mathrm {O}\left (x^{8}\right )\right ) c_{2}+\left (\ln \relax (x ) c_{2}+c_{1}\right ) \left (1-\frac {1}{4} x^{2}-\frac {1}{9} x^{3}+\frac {1}{64} x^{4}+\frac {13}{900} x^{5}+\frac {55}{20736} x^{6}-\frac {433}{705600} x^{7}+\mathrm {O}\left (x^{8}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.005 (sec). Leaf size: 144
AsymptoticDSolveValue[x*y''[x]+y'[x]+x*(1+x)*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 \left (-\frac {433 x^7}{705600}+\frac {55 x^6}{20736}+\frac {13 x^5}{900}+\frac {x^4}{64}-\frac {x^3}{9}-\frac {x^2}{4}+1\right )+c_2 \left (\frac {153527 x^7}{148176000}-\frac {95 x^6}{41472}-\frac {253 x^5}{13500}-\frac {3 x^4}{128}+\frac {2 x^3}{27}+\frac {x^2}{4}+\left (-\frac {433 x^7}{705600}+\frac {55 x^6}{20736}+\frac {13 x^5}{900}+\frac {x^4}{64}-\frac {x^3}{9}-\frac {x^2}{4}+1\right ) \log (x)\right ) \]