Internal problem ID [1368]
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: 16.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {25 x^{2} y^{\prime \prime }+x \left (15+x \right ) y^{\prime }+\left (x +1\right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.014 (sec). Leaf size: 69
Order:=6; dsolve(25*x^2*diff(y(x),x$2)+x*(15+x)*diff(y(x),x)+(1+x)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = x^{\frac {1}{5}} \left (\left (\ln \relax (x ) c_{2}+c_{1}\right ) \left (1-\frac {6}{125} x +\frac {33}{31250} x^{2}-\frac {88}{5859375} x^{3}+\frac {77}{488281250} x^{4}-\frac {1001}{762939453125} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+\left (\frac {7}{125} x -\frac {113}{62500} x^{2}+\frac {1091}{35156250} x^{3}-\frac {1721}{4687500000} x^{4}+\frac {609221}{183105468750000} x^{5}+\mathrm {O}\left (x^{6}\right )\right ) c_{2}\right ) \]
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 134
AsymptoticDSolveValue[25*x^2*y''[x]+x*(15+x)*y'[x]+(1+x)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \sqrt [5]{x} \left (-\frac {1001 x^5}{762939453125}+\frac {77 x^4}{488281250}-\frac {88 x^3}{5859375}+\frac {33 x^2}{31250}-\frac {6 x}{125}+1\right )+c_2 \left (\sqrt [5]{x} \left (\frac {609221 x^5}{183105468750000}-\frac {1721 x^4}{4687500000}+\frac {1091 x^3}{35156250}-\frac {113 x^2}{62500}+\frac {7 x}{125}\right )+\sqrt [5]{x} \left (-\frac {1001 x^5}{762939453125}+\frac {77 x^4}{488281250}-\frac {88 x^3}{5859375}+\frac {33 x^2}{31250}-\frac {6 x}{125}+1\right ) \log (x)\right ) \]