46.2.13 problem 15

Internal problem ID [7316]
Book : ADVANCED ENGINEERING MATHEMATICS. ERWIN KREYSZIG, HERBERT KREYSZIG, EDWARD J. NORMINTON. 10th edition. John Wiley USA. 2011
Section : Chapter 5. Series Solutions of ODEs. Special Functions. Problem set 5.3. Extended Power Series Method: Frobenius Method page 186
Problem number : 15
Date solved : Monday, January 27, 2025 at 02:50:13 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} 2 x \left (1-x \right ) y^{\prime \prime }-\left (1+6 x \right ) y^{\prime }-2 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: 44

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

\[ y = c_{1} x^{{3}/{2}} \left (1+\frac {5}{2} x +\frac {35}{8} x^{2}+\frac {105}{16} x^{3}+\frac {1155}{128} x^{4}+\frac {3003}{256} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (1-2 x -8 x^{2}-16 x^{3}-\frac {128}{5} x^{4}-\frac {256}{7} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.008 (sec). Leaf size: 79

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

\[ y(x)\to c_2 \left (-\frac {256 x^5}{7}-\frac {128 x^4}{5}-16 x^3-8 x^2-2 x+1\right )+c_1 \left (\frac {3003 x^5}{256}+\frac {1155 x^4}{128}+\frac {105 x^3}{16}+\frac {35 x^2}{8}+\frac {5 x}{2}+1\right ) x^{3/2} \]