13.15.1 problem 1

Internal problem ID [2468]
Book : Differential equations and their applications, 3rd ed., M. Braun
Section : Section 2.8.3, The method of Frobenius. Equal roots, and roots differering by an integer. Page 223
Problem number : 1
Date solved : Monday, January 27, 2025 at 05:53:29 AM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} t y^{\prime \prime }+y^{\prime }-4 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(t*diff(y(t),t$2)+diff(y(t),t)-4*y(t)=0,y(t),type='series',t=0);
 

\[ y = \left (c_2 \ln \left (t \right )+c_1 \right ) \left (1+4 t +4 t^{2}+\frac {16}{9} t^{3}+\frac {4}{9} t^{4}+\frac {16}{225} t^{5}+\operatorname {O}\left (t^{6}\right )\right )+\left (\left (-8\right ) t -12 t^{2}-\frac {176}{27} t^{3}-\frac {50}{27} t^{4}-\frac {1096}{3375} t^{5}+\operatorname {O}\left (t^{6}\right )\right ) c_2 \]

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 105

AsymptoticDSolveValue[t*D[y[t],{t,2}]+D[y[t],t]-4*y[t]==0,y[t],{t,0,"6"-1}]
 

\[ y(t)\to c_1 \left (\frac {16 t^5}{225}+\frac {4 t^4}{9}+\frac {16 t^3}{9}+4 t^2+4 t+1\right )+c_2 \left (-\frac {1096 t^5}{3375}-\frac {50 t^4}{27}-\frac {176 t^3}{27}-12 t^2+\left (\frac {16 t^5}{225}+\frac {4 t^4}{9}+\frac {16 t^3}{9}+4 t^2+4 t+1\right ) \log (t)-8 t\right ) \]