Internal problem ID [1353]
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: 1.
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 (1-x \right ) y^{\prime }+\left (-x^{2}+1\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: 75
Order:=8; dsolve(x^2*diff(y(x),x$2)-x*(1-x)*diff(y(x),x)+(1-x^2)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \left (\left (c_{2} \ln \relax (x )+c_{1}\right ) \left (1-x +\frac {3}{4} x^{2}-\frac {13}{36} x^{3}+\frac {79}{576} x^{4}-\frac {67}{1600} x^{5}+\frac {5593}{518400} x^{6}-\frac {60859}{25401600} x^{7}+\mathrm {O}\left (x^{8}\right )\right )+\left (x -x^{2}+\frac {65}{108} x^{3}-\frac {895}{3456} x^{4}+\frac {12547}{144000} x^{5}-\frac {41729}{1728000} x^{6}+\frac {10121677}{1778112000} x^{7}+\mathrm {O}\left (x^{8}\right )\right ) c_{2}\right ) x \]
✓ Solution by Mathematica
Time used: 0.008 (sec). Leaf size: 154
AsymptoticDSolveValue[x^2*y''[x]-x*(1-x)*y'[x]+(1-x^2)*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 x \left (-\frac {60859 x^7}{25401600}+\frac {5593 x^6}{518400}-\frac {67 x^5}{1600}+\frac {79 x^4}{576}-\frac {13 x^3}{36}+\frac {3 x^2}{4}-x+1\right )+c_2 \left (x \left (\frac {10121677 x^7}{1778112000}-\frac {41729 x^6}{1728000}+\frac {12547 x^5}{144000}-\frac {895 x^4}{3456}+\frac {65 x^3}{108}-x^2+x\right )+x \left (-\frac {60859 x^7}{25401600}+\frac {5593 x^6}{518400}-\frac {67 x^5}{1600}+\frac {79 x^4}{576}-\frac {13 x^3}{36}+\frac {3 x^2}{4}-x+1\right ) \log (x)\right ) \]