46.2.14 problem 16

Internal problem ID [7317]
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 : 16
Date solved : Monday, January 27, 2025 at 02:50:15 PM
CAS classification : [_Jacobi]

\begin{align*} x \left (1-x \right ) y^{\prime \prime }+\left (\frac {1}{2}+2 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: 0.012 (sec). Leaf size: 36

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

\[ y = c_{1} \sqrt {x}\, \left (1+\frac {1}{2} x -\frac {1}{40} x^{2}-\frac {1}{560} x^{3}-\frac {1}{2688} x^{4}-\frac {1}{8448} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (1+4 x +\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.009 (sec). Leaf size: 55

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

\[ y(x)\to c_1 \sqrt {x} \left (-\frac {x^5}{8448}-\frac {x^4}{2688}-\frac {x^3}{560}-\frac {x^2}{40}+\frac {x}{2}+1\right )+c_2 (4 x+1) \]