13.14.9 problem 9

Internal problem ID [2449]
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 : 9
Date solved : Monday, January 27, 2025 at 05:53:06 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 t y^{\prime \prime }+\left (1+t \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: 38

Order:=6; 
dsolve(2*t*diff(y(t),t$2)+(1+t)*diff(y(t),t)-2*y(t)=0,y(t),type='series',t=0);
 

\[ y = c_1 \sqrt {t}\, \left (1+\frac {1}{2} t +\frac {1}{40} t^{2}-\frac {1}{1680} t^{3}+\frac {1}{40320} t^{4}-\frac {1}{887040} t^{5}+\operatorname {O}\left (t^{6}\right )\right )+c_2 \left (1+2 t +\frac {1}{3} t^{2}+\operatorname {O}\left (t^{6}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 62

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

\[ y(t)\to c_2 \left (\frac {t^2}{3}+2 t+1\right )+c_1 \sqrt {t} \left (-\frac {t^5}{887040}+\frac {t^4}{40320}-\frac {t^3}{1680}+\frac {t^2}{40}+\frac {t}{2}+1\right ) \]