12.11.10 problem 21

Internal problem ID [1849]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.1 Exercises. Page 318
Problem number : 21
Date solved : Monday, January 27, 2025 at 05:37:12 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (1-x \right ) y^{\prime \prime }+x \left (x +4\right ) y^{\prime }+\left (2-x \right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.012 (sec). Leaf size: 50

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

\[ y = \frac {\ln \left (x \right ) \left (9 x +18 x^{2}+3 x^{3}+\operatorname {O}\left (x^{6}\right )\right ) c_2 +c_1 \left (1+2 x +\frac {1}{3} x^{2}+\operatorname {O}\left (x^{6}\right )\right ) x +\left (1-5 x -55 x^{2}-\frac {53}{3} x^{3}+\operatorname {O}\left (x^{6}\right )\right ) c_2}{x^{2}} \]

✓ Solution by Mathematica

Time used: 0.054 (sec). Leaf size: 56

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

\[ y(x)\to c_1 \left (\frac {3 \left (x^2+6 x+3\right ) \log (x)}{x}-\frac {21 x^3+75 x^2+15 x-1}{x^2}\right )+c_2 \left (\frac {x}{3}+\frac {1}{x}+2\right ) \]