12.16.29 problem 25

Internal problem ID [2091]
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 : 25
Date solved : Monday, January 27, 2025 at 05:42:06 AM
CAS classification : [_Lienard]

\begin{align*} x y^{\prime \prime }-5 y^{\prime }+y x&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.009 (sec). Leaf size: 32

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

\[ y = c_1 \,x^{6} \left (1-\frac {1}{16} x^{2}+\frac {1}{640} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (-86400-10800 x^{2}-1350 x^{4}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.008 (sec). Leaf size: 44

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

\[ y(x)\to c_1 \left (\frac {x^4}{64}+\frac {x^2}{8}+1\right )+c_2 \left (\frac {x^{10}}{640}-\frac {x^8}{16}+x^6\right ) \]