12.16.1 problem Example 7.7.1 page 381

Internal problem ID [2063]
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 : Example 7.7.1 page 381
Date solved : Monday, January 27, 2025 at 05:41:22 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 x^{2} \left (x +2\right ) y^{\prime \prime }-x \left (4-7 x \right ) y^{\prime }-\left (5-3 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: 62

Order:=6; 
dsolve(2*x^2*(2+x)*diff(y(x),x$2)-x*(4-7*x)*diff(y(x),x)-(5-3*x)*y(x)=0,y(x),type='series',x=0);
 

\[ y = \frac {c_1 \,x^{3} \left (1-\frac {7}{4} x +\frac {63}{32} x^{2}-\frac {231}{128} x^{3}+\frac {3003}{2048} x^{4}-\frac {9009}{8192} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (\ln \left (x \right ) \left (-\frac {45}{32} x^{3}+\frac {315}{128} x^{4}-\frac {2835}{1024} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (12+\frac {3}{2} x +\frac {9}{8} x^{2}-\frac {981}{64} x^{3}+\frac {6417}{256} x^{4}-\frac {28089}{1024} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )}{\sqrt {x}} \]

✓ Solution by Mathematica

Time used: 0.053 (sec). Leaf size: 98

AsymptoticDSolveValue[2*x^2*(2+x)*D[y[x],{x,2}]-x*(4-7*x)*D[y[x],x]-(5-3*x)*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_2 \left (\frac {3003 x^{13/2}}{2048}-\frac {231 x^{11/2}}{128}+\frac {63 x^{9/2}}{32}-\frac {7 x^{7/2}}{4}+x^{5/2}\right )+c_1 \left (\frac {15}{512} (7 x-4) x^{5/2} \log (x)+\frac {809 x^4-548 x^3+96 x^2+128 x+1024}{1024 \sqrt {x}}\right ) \]