12.16.17 problem 13

Internal problem ID [2079]
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 : 13
Date solved : Monday, January 27, 2025 at 05:41:48 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} x \left (1+x \right ) y^{\prime \prime }-4 y^{\prime }-2 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: 40

Order:=6; 
dsolve(x*(1+x)*diff(y(x),x$2)-4*diff(y(x),x)-2*y(x)=0,y(x),type='series',x=0);
 

\[ y = c_1 \,x^{5} \left (1-3 x +6 x^{2}-10 x^{3}+15 x^{4}-21 x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (2880-1440 x +480 x^{2}-480 x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.039 (sec). Leaf size: 48

AsymptoticDSolveValue[x*(1+x)*D[y[x],{x,2}]-4*D[y[x],x]-2*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_1 \left (\frac {x^2}{6}-\frac {x}{2}+1\right )+c_2 \left (15 x^9-10 x^8+6 x^7-3 x^6+x^5\right ) \]