Internal problem ID [1371]
Book: Elementary differential equations with boundary value problems. William F. Trench.
Brooks/Cole 2001
Section: Chapter 7 Series Solutions of Linear Second Equations. 7.6 THE METHOD OF FROBENIUS
II. Exercises 7.6. Page 374
Problem number: 19.
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 (-2 x +3\right ) y^{\prime }+\left (3 x +4\right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.009 (sec). Leaf size: 69
Order:=6; dsolve(x^2*diff(y(x),x$2)-x*(3-2*x)*diff(y(x),x)+(4+3*x)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \left (\left (\ln \relax (x ) c_{2}+c_{1}\right ) \left (1-7 x +\frac {63}{4} x^{2}-\frac {77}{4} x^{3}+\frac {1001}{64} x^{4}-\frac {3003}{320} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+\left (12 x -\frac {157}{4} x^{2}+\frac {2063}{36} x^{3}-\frac {59875}{1152} x^{4}+\frac {323399}{9600} x^{5}+\mathrm {O}\left (x^{6}\right )\right ) c_{2}\right ) x^{2} \]
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 122
AsymptoticDSolveValue[x^2*y''[x]-x*(3-2*x)*y'[x]+(4+3*x)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (-\frac {3003 x^5}{320}+\frac {1001 x^4}{64}-\frac {77 x^3}{4}+\frac {63 x^2}{4}-7 x+1\right ) x^2+c_2 \left (\left (\frac {323399 x^5}{9600}-\frac {59875 x^4}{1152}+\frac {2063 x^3}{36}-\frac {157 x^2}{4}+12 x\right ) x^2+\left (-\frac {3003 x^5}{320}+\frac {1001 x^4}{64}-\frac {77 x^3}{4}+\frac {63 x^2}{4}-7 x+1\right ) x^2 \log (x)\right ) \]