12.15.50 problem 46

Internal problem ID [2048]
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 : 46
Date solved : Monday, January 27, 2025 at 05:41:04 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} x^{2} \left (1-x \right ) y^{\prime \prime }+x \left (3-2 x \right ) y^{\prime }+\left (2 x +1\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: 42

Order:=6; 
dsolve(x^2*(1-x)*diff(y(x),x$2)+x*(3-2*x)*diff(y(x),x)+(1+2*x)*y(x)=0,y(x),type='series',x=0);
 

\[ y = \frac {\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1-2 x +x^{2}+\operatorname {O}\left (x^{6}\right )\right )+\left (3 x -3 x^{2}+\frac {1}{3} x^{3}+\frac {1}{12} x^{4}+\frac {1}{30} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2}{x} \]

✓ Solution by Mathematica

Time used: 0.008 (sec). Leaf size: 70

AsymptoticDSolveValue[x^2*(1-x)*D[y[x],{x,2}]+x*(3-2*x)*D[y[x],x]+(1+2*x)*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to \frac {c_1 \left (x^2-2 x+1\right )}{x}+c_2 \left (\frac {\left (x^2-2 x+1\right ) \log (x)}{x}+\frac {\frac {x^5}{30}+\frac {x^4}{12}+\frac {x^3}{3}-3 x^2+3 x}{x}\right ) \]