12.16.24 problem 20

Internal problem ID [2086]
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 : 20
Date solved : Monday, January 27, 2025 at 05:41:58 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (1+x \right ) y^{\prime \prime }-x \left (6+11 x \right ) y^{\prime }+\left (6+32 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: 37

Order:=6; 
dsolve(x^2*(1+x)*diff(y(x),x$2)-x*(6+11*x)*diff(y(x),x)+(6+32*x)*y(x)=0,y(x),type='series',x=0);
 

\[ y = c_1 \,x^{6} \left (1+\frac {2}{3} x +\frac {1}{7} x^{2}+\operatorname {O}\left (x^{6}\right )\right )+c_2 x \left (2880+15120 x +30240 x^{2}+25200 x^{3}-15120 x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.048 (sec). Leaf size: 51

AsymptoticDSolveValue[x^2*(1+x)*D[y[x],{x,2}]-x*(6+11*x)*D[y[x],x]+(6+32*x)*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_2 \left (\frac {x^8}{7}+\frac {2 x^7}{3}+x^6\right )+c_1 \left (\frac {35 x^4}{4}+\frac {21 x^3}{2}+\frac {21 x^2}{4}+x\right ) \]