12.16.23 problem 19

Internal problem ID [2085]
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 : 19
Date solved : Monday, January 27, 2025 at 05:41:57 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (2 x +1\right ) y^{\prime \prime }-2 x \left (3+14 x \right ) y^{\prime }+\left (6+100 x \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: 43

Order:=6; 
dsolve(x^2*(1+2*x)*diff(y(x),x$2)-2*x*(3+14*x)*diff(y(x),x)+(6+100*x)*y(x)=0,y(x),type='series',x=0);
 

\[ y = c_1 \,x^{6} \left (1+\frac {4}{3} x +\frac {8}{7} x^{2}+\frac {4}{7} x^{3}+\frac {8}{63} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_2 x \left (2880+51840 x +414720 x^{2}+1935360 x^{3}+5806080 x^{4}[phone] x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.048 (sec). Leaf size: 64

AsymptoticDSolveValue[x^2*(1+2*x)*D[y[x],{x,2}]-2*x*(3+14*x)*D[y[x],x]+(6+100*x)*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_1 \left (2016 x^5+672 x^4+144 x^3+18 x^2+x\right )+c_2 \left (\frac {8 x^{10}}{63}+\frac {4 x^9}{7}+\frac {8 x^8}{7}+\frac {4 x^7}{3}+x^6\right ) \]