Internal problem ID [1293]
Book: Elementary differential equations with boundary value problems. William F. Trench.
Brooks/Cole 2001
Section: Chapter 7 Series Solutions of Linear Second Equations. 7.5 THE METHOD OF FROBENIUS
I. Exercises 7.5. Page 358
Problem number: Example 7.5.2 page 354.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _exact, _linear, _homogeneous]]
Solve \begin {gather*} \boxed {x^{2} \left (x +3\right ) y^{\prime \prime }+5 x \left (x +1\right ) y^{\prime }-\left (-4 x +1\right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 47
Order:=6; dsolve(x^2*(3+x)*diff(y(x),x$2)+5*x*(1+x)*diff(y(x),x)-(1-4*x)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {c_{2} x^{\frac {4}{3}} \left (1-\frac {7}{9} x +\frac {35}{81} x^{2}-\frac {455}{2187} x^{3}+\frac {1820}{19683} x^{4}-\frac {6916}{177147} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+c_{1} \left (1+x -x^{2}+\frac {3}{5} x^{3}-\frac {3}{10} x^{4}+\frac {3}{22} x^{5}+\mathrm {O}\left (x^{6}\right )\right )}{x} \]
✓ Solution by Mathematica
Time used: 0.006 (sec). Leaf size: 82
AsymptoticDSolveValue[x^2*(3+x)*y''[x]+5*x*(1+x)*y'[x]-(1-4*x)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \sqrt [3]{x} \left (-\frac {6916 x^5}{177147}+\frac {1820 x^4}{19683}-\frac {455 x^3}{2187}+\frac {35 x^2}{81}-\frac {7 x}{9}+1\right )+\frac {c_2 \left (\frac {3 x^5}{22}-\frac {3 x^4}{10}+\frac {3 x^3}{5}-x^2+x+1\right )}{x} \]