Internal problem ID [1365]
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: 13.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {36 x^{2} \left (1-2 x \right ) y^{\prime \prime }+24 x \left (1-9 x \right ) y^{\prime }+\left (1-70 x \right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.018 (sec). Leaf size: 69
Order:=6; dsolve(36*x^2*(1-2*x)*diff(y(x),x$2)+24*x*(1-9*x)*diff(y(x),x)+(1-70*x)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = x^{\frac {1}{6}} \left (\left (\ln \relax (x ) c_{2}+c_{1}\right ) \left (1+\frac {8}{3} x +\frac {56}{9} x^{2}+\frac {1120}{81} x^{3}+\frac {7280}{243} x^{4}+\frac {46592}{729} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+\left (-\frac {2}{3} x -2 x^{2}-\frac {1192}{243} x^{3}-\frac {8168}{729} x^{4}-\frac {270112}{10935} x^{5}+\mathrm {O}\left (x^{6}\right )\right ) c_{2}\right ) \]
✓ Solution by Mathematica
Time used: 0.008 (sec). Leaf size: 132
AsymptoticDSolveValue[36*x^2*(1-2*x)*y''[x]+24*x*(1-9*x)*y'[x]+(1-70*x)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \sqrt [6]{x} \left (\frac {46592 x^5}{729}+\frac {7280 x^4}{243}+\frac {1120 x^3}{81}+\frac {56 x^2}{9}+\frac {8 x}{3}+1\right )+c_2 \left (\sqrt [6]{x} \left (-\frac {270112 x^5}{10935}-\frac {8168 x^4}{729}-\frac {1192 x^3}{243}-2 x^2-\frac {2 x}{3}\right )+\sqrt [6]{x} \left (\frac {46592 x^5}{729}+\frac {7280 x^4}{243}+\frac {1120 x^3}{81}+\frac {56 x^2}{9}+\frac {8 x}{3}+1\right ) \log (x)\right ) \]