7.16.4 problem 4

Internal problem ID [501]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 3. Power series methods. Section 3.4 (Method of Frobenius: The exceptional cases). Problems at page 246
Problem number : 4
Date solved : Monday, January 27, 2025 at 02:54:11 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} 5 x y^{\prime \prime }+\left (30+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: 0.007 (sec). Leaf size: 44

Order:=6; 
dsolve(5*x*diff(y(x),x$2)+(30+3*x)*diff(y(x),x)+3*y(x)=0,y(x),type='series',x=0);
 

\[ y = c_1 \left (1-\frac {1}{10} x +\frac {3}{350} x^{2}-\frac {9}{14000} x^{3}+\frac {3}{70000} x^{4}-\frac {9}{3500000} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_2 \left (2880-1728 x +\frac {2592}{5} x^{2}-\frac {2592}{25} x^{3}+\frac {1944}{125} x^{4}-\frac {5832}{3125} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{5}} \]

✓ Solution by Mathematica

Time used: 0.026 (sec). Leaf size: 70

AsymptoticDSolveValue[5*x*D[y[x],{x,2}]+(30+3*x)*D[y[x],x]+3*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_2 \left (\frac {3 x^4}{70000}-\frac {9 x^3}{14000}+\frac {3 x^2}{350}-\frac {x}{10}+1\right )+c_1 \left (\frac {1}{x^5}-\frac {3}{5 x^4}+\frac {9}{50 x^3}-\frac {9}{250 x^2}+\frac {27}{5000 x}\right ) \]