13.14.8 problem 8

Internal problem ID [2448]
Book : Differential equations and their applications, 3rd ed., M. Braun
Section : Section 2.8.2, Regular singular points, the method of Frobenius. Page 214
Problem number : 8
Date solved : Monday, January 27, 2025 at 05:53:04 AM
CAS classification : [_Laguerre]

\begin{align*} 2 t y^{\prime \prime }+\left (1-2 t \right ) y^{\prime }-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*t*diff(y(t),t$2)+(1-2*t)*diff(y(t),t)-y(t)=0,y(t),type='series',t=0);
 

\[ y = c_1 \sqrt {t}\, \left (1+\frac {2}{3} t +\frac {4}{15} t^{2}+\frac {8}{105} t^{3}+\frac {16}{945} t^{4}+\frac {32}{10395} t^{5}+\operatorname {O}\left (t^{6}\right )\right )+c_2 \left (1+t +\frac {1}{2} t^{2}+\frac {1}{6} t^{3}+\frac {1}{24} t^{4}+\frac {1}{120} t^{5}+\operatorname {O}\left (t^{6}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 81

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

\[ y(t)\to c_1 \sqrt {t} \left (\frac {32 t^5}{10395}+\frac {16 t^4}{945}+\frac {8 t^3}{105}+\frac {4 t^2}{15}+\frac {2 t}{3}+1\right )+c_2 \left (\frac {t^5}{120}+\frac {t^4}{24}+\frac {t^3}{6}+\frac {t^2}{2}+t+1\right ) \]