Internal problem ID [1399]
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: 47.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {4 x^{2} \left (x +1\right ) y^{\prime \prime }+4 y^{\prime } x^{2}+\left (1-5 x \right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 69
Order:=6; dsolve(4*x^2*(1+x)*diff(y(x),x$2)+4*x^2*diff(y(x),x)+(1-5*x)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \left (\left (c_{2} \ln \relax (x )+c_{1}\right ) \left (1+x -\frac {1}{4} x^{2}+\frac {5}{36} x^{3}-\frac {55}{576} x^{4}+\frac {209}{2880} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+\left (\left (-3\right ) x +\frac {1}{4} x^{2}-\frac {5}{54} x^{3}+\frac {175}{3456} x^{4}-\frac {2863}{86400} x^{5}+\mathrm {O}\left (x^{6}\right )\right ) c_{2}\right ) \sqrt {x} \]
✓ Solution by Mathematica
Time used: 0.007 (sec). Leaf size: 124
AsymptoticDSolveValue[4*x^2*(1+x)*y''[x]+4*x^2*y'[x]+(1-5*x)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \sqrt {x} \left (\frac {209 x^5}{2880}-\frac {55 x^4}{576}+\frac {5 x^3}{36}-\frac {x^2}{4}+x+1\right )+c_2 \left (\sqrt {x} \left (-\frac {2863 x^5}{86400}+\frac {175 x^4}{3456}-\frac {5 x^3}{54}+\frac {x^2}{4}-3 x\right )+\sqrt {x} \left (\frac {209 x^5}{2880}-\frac {55 x^4}{576}+\frac {5 x^3}{36}-\frac {x^2}{4}+x+1\right ) \log (x)\right ) \]