12.16.22 problem 18

Internal problem ID [2084]
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 : 18
Date solved : Monday, January 27, 2025 at 05:41:55 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (1+x \right ) y^{\prime \prime }+3 x^{2} y^{\prime }-\left (6-x \right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.009 (sec). Leaf size: 39

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

\[ y = c_1 \,x^{3} \left (1-\frac {8}{3} x +\frac {100}{21} x^{2}-\frac {50}{7} x^{3}+\frac {175}{18} x^{4}-\frac {112}{9} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_2 \left (2880+720 x +\operatorname {O}\left (x^{6}\right )\right )}{x^{2}} \]

✓ Solution by Mathematica

Time used: 0.055 (sec). Leaf size: 53

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

\[ y(x)\to c_1 \left (\frac {1}{x^2}+\frac {1}{4 x}\right )+c_2 \left (\frac {175 x^7}{18}-\frac {50 x^6}{7}+\frac {100 x^5}{21}-\frac {8 x^4}{3}+x^3\right ) \]