54.6.12 problem 12

Internal problem ID [8644]
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 : 12
Date solved : Monday, January 27, 2025 at 04:18:10 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x y^{\prime \prime }+\left (2 x +3\right ) y^{\prime }+4 y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Solution by Maple

Time used: 0.017 (sec). Leaf size: 50

Order:=8; 
dsolve(x*diff(y(x),x$2)+(3+2*x)*diff(y(x),x)+4*y(x)=0,y(x),type='series',x=0);
 

\[ y = c_{1} \left (1-\frac {4}{3} x +x^{2}-\frac {8}{15} x^{3}+\frac {2}{9} x^{4}-\frac {8}{105} x^{5}+\frac {1}{45} x^{6}-\frac {16}{2835} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\frac {c_{2} \left (-2+4 x^{2}-\frac {16}{3} x^{3}+4 x^{4}-\frac {32}{15} x^{5}+\frac {8}{9} x^{6}-\frac {32}{105} x^{7}+\operatorname {O}\left (x^{8}\right )\right )}{x^{2}} \]

✓ Solution by Mathematica

Time used: 0.099 (sec). Leaf size: 77

AsymptoticDSolveValue[x*D[y[x],{x,2}]+(3+2*x)*D[y[x],x]+4*y[x]==0,y[x],{x,0,"8"-1}]
 

\[ y(x)\to c_1 \left (-\frac {4 x^4}{9}+\frac {16 x^3}{15}-2 x^2+\frac {1}{x^2}+\frac {8 x}{3}-2\right )+c_2 \left (\frac {x^6}{45}-\frac {8 x^5}{105}+\frac {2 x^4}{9}-\frac {8 x^3}{15}+x^2-\frac {4 x}{3}+1\right ) \]