13.14.20 problem 20

Internal problem ID [2460]
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 : 20
Date solved : Monday, January 27, 2025 at 05:53:18 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 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: 0.009 (sec). Leaf size: 62

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

\[ y = \frac {c_1 \,t^{2} \left (1+\frac {1}{3} t +\frac {1}{24} t^{2}+\frac {1}{360} t^{3}+\frac {1}{8640} t^{4}+\frac {1}{302400} t^{5}+\operatorname {O}\left (t^{6}\right )\right )+c_2 \left (\ln \left (t \right ) \left (t^{2}+\frac {1}{3} t^{3}+\frac {1}{24} t^{4}+\frac {1}{360} t^{5}+\operatorname {O}\left (t^{6}\right )\right )+\left (-2+2 t -\frac {4}{9} t^{3}-\frac {25}{288} t^{4}-\frac {157}{21600} t^{5}+\operatorname {O}\left (t^{6}\right )\right )\right )}{t} \]

✓ Solution by Mathematica

Time used: 0.021 (sec). Leaf size: 83

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

\[ y(t)\to c_1 \left (\frac {31 t^4+176 t^3+144 t^2-576 t+576}{576 t}-\frac {1}{48} t \left (t^2+8 t+24\right ) \log (t)\right )+c_2 \left (\frac {t^5}{8640}+\frac {t^4}{360}+\frac {t^3}{24}+\frac {t^2}{3}+t\right ) \]