Internal problem ID [1350]
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: Example 7.6.2 page 369.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {2 x^{2} \left (2+x \right ) y^{\prime \prime }+5 y^{\prime } x^{2}+\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: 0.013 (sec). Leaf size: 69
Order:=6; dsolve(2*x^2*(2+x)*diff(y(x),x$2)+5*x^2*diff(y(x),x)+(1+x)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \sqrt {x}\, \left (\left (\ln \relax (x ) c_{2}+c_{1}\right ) \left (1-\frac {3}{4} x +\frac {15}{32} x^{2}-\frac {35}{128} x^{3}+\frac {315}{2048} x^{4}-\frac {693}{8192} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+\left (\frac {1}{4} x -\frac {13}{64} x^{2}+\frac {101}{768} x^{3}-\frac {641}{8192} x^{4}+\frac {7303}{163840} x^{5}+\mathrm {O}\left (x^{6}\right )\right ) c_{2}\right ) \]
✓ Solution by Mathematica
Time used: 0.008 (sec). Leaf size: 134
AsymptoticDSolveValue[2*x^2*(2+x)*y''[x]+5*x^2*y'[x]+(1+x)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \sqrt {x} \left (-\frac {693 x^5}{8192}+\frac {315 x^4}{2048}-\frac {35 x^3}{128}+\frac {15 x^2}{32}-\frac {3 x}{4}+1\right )+c_2 \left (\sqrt {x} \left (\frac {7303 x^5}{163840}-\frac {641 x^4}{8192}+\frac {101 x^3}{768}-\frac {13 x^2}{64}+\frac {x}{4}\right )+\sqrt {x} \left (-\frac {693 x^5}{8192}+\frac {315 x^4}{2048}-\frac {35 x^3}{128}+\frac {15 x^2}{32}-\frac {3 x}{4}+1\right ) \log (x)\right ) \]