12.14.29 problem 31

Internal problem ID [1970]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.5 THE METHOD OF FROBENIUS I. Exercises 7.5. Page 358
Problem number : 31
Date solved : Monday, January 27, 2025 at 05:39:24 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (x +2\right ) y^{\prime \prime }+5 x \left (1-x \right ) y^{\prime }-\left (2-8 x \right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.009 (sec). Leaf size: 46

Order:=6; 
dsolve(x^2*(2+x)*diff(y(x),x$2)+5*x*(1-x)*diff(y(x),x)-(2-8*x)*y(x)=0,y(x),type='series',x=0);
 

\[ y = \frac {c_2 \,x^{{5}/{2}} \left (1-\frac {3}{4} x +\frac {5}{96} x^{2}+\frac {5}{4224} x^{3}+\frac {5}{292864} x^{4}-\frac {1}{3514368} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_1 \left (1+8 x +60 x^{2}-160 x^{3}+40 x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{2}} \]

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 73

AsymptoticDSolveValue[x^2*(2+x)*D[y[x],{x,2}]+5*x*(1-x)*D[y[x],x]-(2-8*x)*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to \frac {c_2 \left (40 x^4-160 x^3+60 x^2+8 x+1\right )}{x^2}+c_1 \sqrt {x} \left (-\frac {x^5}{3514368}+\frac {5 x^4}{292864}+\frac {5 x^3}{4224}+\frac {5 x^2}{96}-\frac {3 x}{4}+1\right ) \]