54.6.10 problem 10

Internal problem ID [8642]
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 : 10
Date solved : Monday, January 27, 2025 at 04:18:07 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.020 (sec). Leaf size: 52

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

\[ y = c_{1} \left (1-\frac {3}{4} x +\frac {9}{20} x^{2}-\frac {9}{40} x^{3}+\frac {27}{280} x^{4}-\frac {81}{2240} x^{5}+\frac {27}{2240} x^{6}-\frac {81}{22400} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\frac {c_{2} \left (12-36 x +54 x^{2}-54 x^{3}+\frac {81}{2} x^{4}-\frac {243}{10} x^{5}+\frac {243}{20} x^{6}-\frac {729}{140} x^{7}+\operatorname {O}\left (x^{8}\right )\right )}{x^{3}} \]

✓ Solution by Mathematica

Time used: 0.104 (sec). Leaf size: 90

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

\[ y(x)\to c_1 \left (\frac {81 x^3}{80}+\frac {1}{x^3}-\frac {81 x^2}{40}-\frac {3}{x^2}+\frac {27 x}{8}+\frac {9}{2 x}-\frac {9}{2}\right )+c_2 \left (\frac {27 x^6}{2240}-\frac {81 x^5}{2240}+\frac {27 x^4}{280}-\frac {9 x^3}{40}+\frac {9 x^2}{20}-\frac {3 x}{4}+1\right ) \]