54.6.6 problem 6

Internal problem ID [8638]
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
Date solved : Monday, January 27, 2025 at 04:18:01 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x \left (1+x \right ) y^{\prime \prime }+\left (5+x \right ) y^{\prime }-4 y&=0 \end{align*}

Using series method with expansion around

\begin{align*} -1 \end{align*}

✓ Solution by Maple

Time used: 0.020 (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);
 

\[ y = c_{1} \left (x +1\right )^{5} \left (1+\frac {7}{2} \left (x +1\right )+8 \left (x +1\right )^{2}+15 \left (x +1\right )^{3}+25 \left (x +1\right )^{4}+\frac {77}{2} \left (x +1\right )^{5}+56 \left (x +1\right )^{6}+78 \left (x +1\right )^{7}+\operatorname {O}\left (\left (x +1\right )^{8}\right )\right )+c_{2} \left (2880+2880 \left (x +1\right )+1440 \left (x +1\right )^{2}+2880 \left (x +1\right )^{5}+10080 \left (x +1\right )^{6}+23040 \left (x +1\right )^{7}+\operatorname {O}\left (\left (x +1\right )^{8}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.121 (sec). Leaf size: 88

AsymptoticDSolveValue[x*(1+x)*D[y[x],{x,2}]+(x+5)*D[y[x],x]-4*y[x]==0,y[x],{x,-1,"8"-1}]
 

\[ y(x)\to c_1 \left (\frac {7}{2} (x+1)^6+(x+1)^5+\frac {1}{2} (x+1)^2+x+2\right )+c_2 \left (56 (x+1)^{11}+\frac {77}{2} (x+1)^{10}+25 (x+1)^9+15 (x+1)^8+8 (x+1)^7+\frac {7}{2} (x+1)^6+(x+1)^5\right ) \]