Internal problem ID [6200]
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: 6.
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 x +1\right ) y^{\prime }+\left (1-6 x \right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.019 (sec). Leaf size: 75
Order:=8; dsolve(x^2*diff(y(x),x$2)-x*(1+3*x)*diff(y(x),x)+(1-6*x)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = x \left (\left (\ln \relax (x ) c_{2}+c_{1}\right ) \left (1+9 x +27 x^{2}+45 x^{3}+\frac {405}{8} x^{4}+\frac {1701}{40} x^{5}+\frac {567}{20} x^{6}+\frac {2187}{140} x^{7}+\mathrm {O}\left (x^{8}\right )\right )+\left (\left (-15\right ) x -\frac {261}{4} x^{2}-\frac {519}{4} x^{3}-\frac {5211}{32} x^{4}-\frac {118179}{800} x^{5}-\frac {83511}{800} x^{6}-\frac {2361717}{39200} x^{7}+\mathrm {O}\left (x^{8}\right )\right ) c_{2}\right ) \]
✓ Solution by Mathematica
Time used: 0.005 (sec). Leaf size: 150
AsymptoticDSolveValue[x^2*y''[x]-x*(1+3*x)*y'[x]+(1-6*x)*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 x \left (\frac {2187 x^7}{140}+\frac {567 x^6}{20}+\frac {1701 x^5}{40}+\frac {405 x^4}{8}+45 x^3+27 x^2+9 x+1\right )+c_2 \left (x \left (-\frac {2361717 x^7}{39200}-\frac {83511 x^6}{800}-\frac {118179 x^5}{800}-\frac {5211 x^4}{32}-\frac {519 x^3}{4}-\frac {261 x^2}{4}-15 x\right )+x \left (\frac {2187 x^7}{140}+\frac {567 x^6}{20}+\frac {1701 x^5}{40}+\frac {405 x^4}{8}+45 x^3+27 x^2+9 x+1\right ) \log (x)\right ) \]