Internal problem ID [1352]
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: Example 7.6.4 page 372.
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 (5-x \right ) y^{\prime }+\left (9-4 x \right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.01 (sec). Leaf size: 53
Order:=6; dsolve(x^2*diff(y(x),x$2)-x*(5-x)*diff(y(x),x)+(9-4*x)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \left (\left (\ln \relax (x ) c_{2}+c_{1}\right ) \left (1+x +\mathrm {O}\left (x^{6}\right )\right )+\left (\left (-3\right ) x -\frac {1}{4} x^{2}+\frac {1}{36} x^{3}-\frac {1}{288} x^{4}+\frac {1}{2400} x^{5}+\mathrm {O}\left (x^{6}\right )\right ) c_{2}\right ) x^{3} \]
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 62
AsymptoticDSolveValue[x^2*y''[x]-x*(5-x)*y'[x]+(9-4*x)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 (x+1) x^3+c_2 \left ((x+1) x^3 \log (x)+\left (\frac {x^5}{2400}-\frac {x^4}{288}+\frac {x^3}{36}-\frac {x^2}{4}-3 x\right ) x^3\right ) \]