Internal problem ID [6198]
Book: Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition.
1997.
Section: CHAPTER 18. Power series solutions. 18.6. Indicial Equation with Equal Roots. Exercises
page 373
Problem number: 4.
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 (4 x^{2}+1\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: 57
Order:=8; dsolve(x^2*diff(y(x),x$2)+3*x*diff(y(x),x)+(1+4*x^2)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {\left (\ln \relax (x ) c_{2}+c_{1}\right ) \left (1-x^{2}+\frac {1}{4} x^{4}-\frac {1}{36} x^{6}+\mathrm {O}\left (x^{8}\right )\right )+\left (x^{2}-\frac {3}{8} x^{4}+\frac {11}{216} x^{6}+\mathrm {O}\left (x^{8}\right )\right ) c_{2}}{x} \]
✓ Solution by Mathematica
Time used: 0.005 (sec). Leaf size: 84
AsymptoticDSolveValue[x^2*y''[x]+3*x*y'[x]+(1+4*x^2)*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to \frac {c_1 \left (-\frac {x^6}{36}+\frac {x^4}{4}-x^2+1\right )}{x}+c_2 \left (\frac {\frac {11 x^6}{216}-\frac {3 x^4}{8}+x^2}{x}+\frac {\left (-\frac {x^6}{36}+\frac {x^4}{4}-x^2+1\right ) \log (x)}{x}\right ) \]