Internal problem ID [1400]
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: 48.
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-x \right ) y^{\prime \prime }-x \left (3-5 x \right ) y^{\prime }+\left (4-5 x \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: 61
Order:=6; dsolve(x^2*(1-x)*diff(y(x),x$2)-x*(3-5*x)*diff(y(x),x)+(4-5*x)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \left (\left (c_{2} \ln \relax (x )+c_{1}\right ) \left (1-3 x +3 x^{2}-x^{3}+\mathrm {O}\left (x^{6}\right )\right )+\left (4 x -7 x^{2}+\frac {11}{3} x^{3}-\frac {1}{4} x^{4}-\frac {1}{20} x^{5}+\mathrm {O}\left (x^{6}\right )\right ) c_{2}\right ) x^{2} \]
✓ Solution by Mathematica
Time used: 0.007 (sec). Leaf size: 84
AsymptoticDSolveValue[x^2*(1-x)*y''[x]-x*(3-5*x)*y'[x]+(4-5*x)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (-x^3+3 x^2-3 x+1\right ) x^2+c_2 \left (\left (-x^3+3 x^2-3 x+1\right ) x^2 \log (x)+\left (-\frac {x^5}{20}-\frac {x^4}{4}+\frac {11 x^3}{3}-7 x^2+4 x\right ) x^2\right ) \]