40.17.2 problem 12

Internal problem ID [6803]
Book : Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres. McGraw Hill 1952
Section : Chapter 26. Integration in series (singular points). Supplemetary problems. Page 218
Problem number : 12
Date solved : Monday, January 27, 2025 at 02:30:48 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.008 (sec). Leaf size: 44

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

\[ y = c_1 \sqrt {x}\, \left (1+\frac {1}{3} x +\frac {1}{15} x^{2}+\frac {1}{105} x^{3}+\frac {1}{945} x^{4}+\frac {1}{10395} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (1+\frac {1}{2} x +\frac {1}{8} x^{2}+\frac {1}{48} x^{3}+\frac {1}{384} x^{4}+\frac {1}{3840} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 85

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

\[ y(x)\to c_1 \sqrt {x} \left (\frac {x^5}{10395}+\frac {x^4}{945}+\frac {x^3}{105}+\frac {x^2}{15}+\frac {x}{3}+1\right )+c_2 \left (\frac {x^5}{3840}+\frac {x^4}{384}+\frac {x^3}{48}+\frac {x^2}{8}+\frac {x}{2}+1\right ) \]