54.6.5 problem 5

Internal problem ID [8637]
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
Date solved : Monday, January 27, 2025 at 04:18:00 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*} 0 \end{align*}

✓ Solution by Maple

Time used: 0.021 (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);
 

\[ y = c_{1} \left (1+\frac {4}{5} x +\frac {1}{5} x^{2}+\operatorname {O}\left (x^{8}\right )\right )+\frac {c_{2} \left (-144-576 x -720 x^{2}+720 x^{4}+576 x^{5}+144 x^{6}+\operatorname {O}\left (x^{8}\right )\right )}{x^{4}} \]

✓ Solution by Mathematica

Time used: 0.122 (sec). Leaf size: 47

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

\[ y(x)\to c_2 \left (\frac {x^2}{5}+\frac {4 x}{5}+1\right )+c_1 \left (\frac {1}{x^4}+\frac {4}{x^3}-x^2+\frac {5}{x^2}-4 x-5\right ) \]