12.16.21 problem 17

Internal problem ID [2083]
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 : 17
Date solved : Monday, January 27, 2025 at 05:41:54 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (1+x \right ) y^{\prime \prime }+x \left (1-10 x \right ) y^{\prime }-\left (9-10 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: 45

Order:=6; 
dsolve(x^2*(1+x)*diff(y(x),x$2)+x*(1-10*x)*diff(y(x),x)-(9-10*x)*y(x)=0,y(x),type='series',x=0);
 

\[ y = c_1 \,x^{3} \left (1+2 x +\frac {9}{4} x^{2}+\frac {5}{3} x^{3}+\frac {5}{6} x^{4}+\frac {3}{11} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_2 \left (-86400-898560 x -4043520 x^{2}-9884160 x^{3}-12355200 x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{3}} \]

✓ Solution by Mathematica

Time used: 0.043 (sec). Leaf size: 64

AsymptoticDSolveValue[x^2*(1+x)*D[y[x],{x,2}]+x*(1-10*x)*D[y[x],x]-(9-10*x)*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_1 \left (\frac {1}{x^3}+\frac {52}{5 x^2}+143 x+\frac {234}{5 x}+\frac {572}{5}\right )+c_2 \left (\frac {5 x^7}{6}+\frac {5 x^6}{3}+\frac {9 x^5}{4}+2 x^4+x^3\right ) \]