Internal problem ID [1376]
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: 24.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{2} \left (1-2 x \right ) y^{\prime \prime }-x \left (4 x +5\right ) y^{\prime }+\left (4 x +9\right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 81
Order:=8; dsolve(x^2*(1-2*x)*diff(y(x),x$2)-x*(5+4*x)*diff(y(x),x)+(9+4*x)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \left (\left (c_{2} \ln \relax (x )+c_{1}\right ) \left (1+20 x +180 x^{2}+1120 x^{3}+5600 x^{4}+24192 x^{5}+94080 x^{6}+337920 x^{7}+\mathrm {O}\left (x^{8}\right )\right )+\left (\left (-26\right ) x -324 x^{2}-\frac {6968}{3} x^{3}-\frac {37780}{3} x^{4}-57360 x^{5}-\frac {694736}{3} x^{6}-\frac {2566144}{3} x^{7}+\mathrm {O}\left (x^{8}\right )\right ) c_{2}\right ) x^{3} \]
✓ Solution by Mathematica
Time used: 0.015 (sec). Leaf size: 136
AsymptoticDSolveValue[x^2*(1-2*x)*y''[x]-x*(5+4*x)*y'[x]+(9+4*x)*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 \left (337920 x^7+94080 x^6+24192 x^5+5600 x^4+1120 x^3+180 x^2+20 x+1\right ) x^3+c_2 \left (\left (-\frac {2566144 x^7}{3}-\frac {694736 x^6}{3}-57360 x^5-\frac {37780 x^4}{3}-\frac {6968 x^3}{3}-324 x^2-26 x\right ) x^3+\left (337920 x^7+94080 x^6+24192 x^5+5600 x^4+1120 x^3+180 x^2+20 x+1\right ) x^3 \log (x)\right ) \]