Internal problem ID [6208]
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: 14.
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 }+x \left (3+2 x \right ) y^{\prime }+\left (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: 0.026 (sec). Leaf size: 79
Order:=8; dsolve(x^2*diff(y(x),x$2)+x*(3+2*x)*diff(y(x),x)+(1+3*x)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {\left (\ln \relax (x ) c_{2}+c_{1}\right ) \left (1-x +\frac {3}{4} x^{2}-\frac {5}{12} x^{3}+\frac {35}{192} x^{4}-\frac {21}{320} x^{5}+\frac {77}{3840} x^{6}-\frac {143}{26880} x^{7}+\mathrm {O}\left (x^{8}\right )\right )+\left (-\frac {1}{4} x^{2}+\frac {1}{4} x^{3}-\frac {19}{128} x^{4}+\frac {25}{384} x^{5}-\frac {317}{13824} x^{6}+\frac {469}{69120} x^{7}+\mathrm {O}\left (x^{8}\right )\right ) c_{2}}{x} \]
✓ Solution by Mathematica
Time used: 0.005 (sec). Leaf size: 161
AsymptoticDSolveValue[x^2*y''[x]+x*(3+2*x)*y'[x]+(1+3*x)*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to \frac {c_1 \left (-\frac {143 x^7}{26880}+\frac {77 x^6}{3840}-\frac {21 x^5}{320}+\frac {35 x^4}{192}-\frac {5 x^3}{12}+\frac {3 x^2}{4}-x+1\right )}{x}+c_2 \left (\frac {\frac {469 x^7}{69120}-\frac {317 x^6}{13824}+\frac {25 x^5}{384}-\frac {19 x^4}{128}+\frac {x^3}{4}-\frac {x^2}{4}}{x}+\frac {\left (-\frac {143 x^7}{26880}+\frac {77 x^6}{3840}-\frac {21 x^5}{320}+\frac {35 x^4}{192}-\frac {5 x^3}{12}+\frac {3 x^2}{4}-x+1\right ) \log (x)}{x}\right ) \]