Internal problem ID [6210]
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: 16.
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 \left (1+x \right ) y^{\prime }+\left (1-3 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: 61
Order:=8; dsolve(x^2*diff(y(x),x$2)+3*x*(1+x)*diff(y(x),x)+(1-3*x)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {\left (\left (-15\right ) x -\frac {81}{4} x^{2}-\frac {3}{2} x^{3}+\frac {9}{32} x^{4}-\frac {27}{400} x^{5}+\frac {27}{1600} x^{6}-\frac {81}{19600} x^{7}+\mathrm {O}\left (x^{8}\right )\right ) c_{2}+\left (\ln \relax (x ) c_{2}+c_{1}\right ) \left (1+6 x +\frac {9}{2} x^{2}+\mathrm {O}\left (x^{8}\right )\right )}{x} \]
✓ Solution by Mathematica
Time used: 0.01 (sec). Leaf size: 94
AsymptoticDSolveValue[x^2*y''[x]+3*x*(1+x)*y'[x]+(1-3*x)*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to \frac {c_1 \left (\frac {9 x^2}{2}+6 x+1\right )}{x}+c_2 \left (\frac {\left (\frac {9 x^2}{2}+6 x+1\right ) \log (x)}{x}+\frac {-\frac {81 x^7}{19600}+\frac {27 x^6}{1600}-\frac {27 x^5}{400}+\frac {9 x^4}{32}-\frac {3 x^3}{2}-\frac {81 x^2}{4}-15 x}{x}\right ) \]