12.16.15 problem 11

Internal problem ID [2077]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.6 THE METHOD OF FROBENIUS III. Exercises 7.7. Page 389
Problem number : 11
Date solved : Monday, January 27, 2025 at 05:41:45 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+x \left (1+x \right ) y^{\prime }-3 \left (x +3\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.010 (sec). Leaf size: 37

Order:=6; 
dsolve(x^2*diff(y(x),x$2)+x*(1+x)*diff(y(x),x)-3*(3+x)*y(x)=0,y(x),type='series',x=0);
 

\[ y = c_1 \,x^{3} \left (1+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_2 \left (-86400+103680 x -64800 x^{2}+28800 x^{3}-10800 x^{4}+4320 x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{3}} \]

✓ Solution by Mathematica

Time used: 0.028 (sec). Leaf size: 39

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

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