34.5.5 problem 10

Internal problem ID [6073]
Book : A treatise on ordinary and partial differential equations by William Woolsey Johnson. 1913
Section : Chapter VII, Solutions in series. Examples XVI. page 220
Problem number : 10
Date solved : Monday, January 27, 2025 at 01:34:54 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} 2 x \left (1-x \right ) y^{\prime \prime }+y^{\prime }+4 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: 38

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

\[ y = c_1 \sqrt {x}\, \left (1-\frac {3}{2} x +\frac {3}{8} x^{2}+\frac {1}{16} x^{3}+\frac {3}{128} x^{4}+\frac {3}{256} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (1-4 x +\frac {8}{3} x^{2}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 62

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

\[ y(x)\to c_2 \left (\frac {8 x^2}{3}-4 x+1\right )+c_1 \sqrt {x} \left (\frac {3 x^5}{256}+\frac {3 x^4}{128}+\frac {x^3}{16}+\frac {3 x^2}{8}-\frac {3 x}{2}+1\right ) \]