Internal problem ID [6217]
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: 6.
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 \(x = -1\).
✓ Solution by Maple
Time used: 0.034 (sec). Leaf size: 50
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=-1);
\[ y \relax (x ) = c_{1} \left (x +1\right )^{5} \left (1+\frac {7}{2} \left (x +1\right )+8 \left (x +1\right )^{2}+15 \left (x +1\right )^{3}+25 \left (x +1\right )^{4}+\frac {77}{2} \left (x +1\right )^{5}+56 \left (x +1\right )^{6}+78 \left (x +1\right )^{7}+\mathrm {O}\left (\left (x +1\right )^{8}\right )\right )+c_{2} \left (2880+2880 \left (x +1\right )+1440 \left (x +1\right )^{2}+2880 \left (x +1\right )^{5}+10080 \left (x +1\right )^{6}+23040 \left (x +1\right )^{7}+\mathrm {O}\left (\left (x +1\right )^{8}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.089 (sec). Leaf size: 88
AsymptoticDSolveValue[x*(1+x)*y''[x]+(x+5)*y'[x]-4*y[x]==0,y[x],{x,-1,7}]
\[ y(x)\to c_1 \left (\frac {7}{2} (x+1)^6+(x+1)^5+\frac {1}{2} (x+1)^2+x+2\right )+c_2 \left (56 (x+1)^{11}+\frac {77}{2} (x+1)^{10}+25 (x+1)^9+15 (x+1)^8+8 (x+1)^7+\frac {7}{2} (x+1)^6+(x+1)^5\right ) \]