12.15.19 problem 15

Internal problem ID [2017]
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 II. Exercises 7.6. Page 374
Problem number : 15
Date solved : Monday, January 27, 2025 at 05:40:25 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (1-2 x \right ) y^{\prime \prime }-x \left (5-4 x \right ) y^{\prime }+\left (9-4 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: 48

Order:=6; 
dsolve(x^2*(1-2*x)*diff(y(x),x$2)-x*(5-4*x)*diff(y(x),x)+(9-4*x)*y(x)=0,y(x),type='series',x=0);
 

\[ y = x^{3} \left (\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1+4 x +12 x^{2}+32 x^{3}+80 x^{4}+192 x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (\left (-2\right ) x -8 x^{2}-24 x^{3}-64 x^{4}-160 x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \right ) \]

✓ Solution by Mathematica

Time used: 0.008 (sec). Leaf size: 98

AsymptoticDSolveValue[x^2*(1-2*x)*D[y[x],{x,2}]-x*(5-4*x)*D[y[x],x]+(9-4*x)*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_1 \left (192 x^5+80 x^4+32 x^3+12 x^2+4 x+1\right ) x^3+c_2 \left (\left (-160 x^5-64 x^4-24 x^3-8 x^2-2 x\right ) x^3+\left (192 x^5+80 x^4+32 x^3+12 x^2+4 x+1\right ) x^3 \log (x)\right ) \]