Internal problem ID [6216]
Book: Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition.
1997.
Section: CHAPTER 18. Power series solutions. 18.8 Indicial Equation with Difference of Roots a
Positive Integer: Nonlogarithmic Case. Exercises page 380
Problem number: 5.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x \left (1+x \right ) y^{\prime \prime }+\left (x +5\right ) y^{\prime }-4 y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.079 (sec). Leaf size: 38
Order:=8; dsolve(x*(1+x)*diff(y(x),x$2)+(x+5)*diff(y(x),x)-4*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} \left (1+\frac {4}{5} x +\frac {1}{5} x^{2}+\mathrm {O}\left (x^{8}\right )\right )+\frac {c_{2} \left (-144-576 x -720 x^{2}+720 x^{4}+576 x^{5}+144 x^{6}+\mathrm {O}\left (x^{8}\right )\right )}{x^{4}} \]
✓ Solution by Mathematica
Time used: 0.199 (sec). Leaf size: 47
AsymptoticDSolveValue[x*(1+x)*y''[x]+(x+5)*y'[x]-4*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_2 \left (\frac {x^2}{5}+\frac {4 x}{5}+1\right )+c_1 \left (\frac {1}{x^4}+\frac {4}{x^3}-x^2+\frac {5}{x^2}-4 x-5\right ) \]