13.14.10 problem 10

Internal problem ID [2450]
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 : 10
Date solved : Monday, January 27, 2025 at 05:53:07 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 t^{2} y^{\prime \prime }-t y^{\prime }+\left (1+t \right ) 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: 45

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

\[ y = c_1 \sqrt {t}\, \left (1-t +\frac {1}{6} t^{2}-\frac {1}{90} t^{3}+\frac {1}{2520} t^{4}-\frac {1}{113400} t^{5}+\operatorname {O}\left (t^{6}\right )\right )+c_2 t \left (1-\frac {1}{3} t +\frac {1}{30} t^{2}-\frac {1}{630} t^{3}+\frac {1}{22680} t^{4}-\frac {1}{1247400} t^{5}+\operatorname {O}\left (t^{6}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 84

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

\[ y(t)\to c_1 t \left (-\frac {t^5}{1247400}+\frac {t^4}{22680}-\frac {t^3}{630}+\frac {t^2}{30}-\frac {t}{3}+1\right )+c_2 \sqrt {t} \left (-\frac {t^5}{113400}+\frac {t^4}{2520}-\frac {t^3}{90}+\frac {t^2}{6}-t+1\right ) \]