13.14.7 problem 7

Internal problem ID [2447]
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 : 7
Date solved : Monday, January 27, 2025 at 05:53:03 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 t^{2} y^{\prime \prime }+3 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: 0.007 (sec). Leaf size: 48

Order:=6; 
dsolve(2*t^2*diff(y(t),t$2)+3*t*diff(y(t),t)-(1+t)*y(t)=0,y(t),type='series',t=0);
 

\[ y = \frac {c_2 \,t^{{3}/{2}} \left (1+\frac {1}{5} t +\frac {1}{70} t^{2}+\frac {1}{1890} t^{3}+\frac {1}{83160} t^{4}+\frac {1}{5405400} t^{5}+\operatorname {O}\left (t^{6}\right )\right )+c_1 \left (1-t -\frac {1}{2} t^{2}-\frac {1}{18} t^{3}-\frac {1}{360} t^{4}-\frac {1}{12600} t^{5}+\operatorname {O}\left (t^{6}\right )\right )}{t} \]

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 86

AsymptoticDSolveValue[2*t^2*D[y[t],{t,2}]+3*t*D[y[t],t]-(1+t)*y[t]==0,y[t],{t,0,"6"-1}]
 

\[ y(t)\to c_1 \sqrt {t} \left (\frac {t^5}{5405400}+\frac {t^4}{83160}+\frac {t^3}{1890}+\frac {t^2}{70}+\frac {t}{5}+1\right )+\frac {c_2 \left (-\frac {t^5}{12600}-\frac {t^4}{360}-\frac {t^3}{18}-\frac {t^2}{2}-t+1\right )}{t} \]