13.15.4 problem 4

Internal problem ID [2471]
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 : 4
Date solved : Monday, January 27, 2025 at 05:53:32 AM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} t y^{\prime \prime }+3 y^{\prime }-3 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: 62

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

\[ y = \frac {c_1 \left (1+t +\frac {3}{8} t^{2}+\frac {3}{40} t^{3}+\frac {3}{320} t^{4}+\frac {9}{11200} t^{5}+\operatorname {O}\left (t^{6}\right )\right ) t^{2}+c_2 \left (\ln \left (t \right ) \left (9 t^{2}+9 t^{3}+\frac {27}{8} t^{4}+\frac {27}{40} t^{5}+\operatorname {O}\left (t^{6}\right )\right )+\left (-2+6 t -12 t^{3}-\frac {225}{32} t^{4}-\frac {1413}{800} t^{5}+\operatorname {O}\left (t^{6}\right )\right )\right )}{t^{2}} \]

✓ Solution by Mathematica

Time used: 0.022 (sec). Leaf size: 78

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

\[ y(t)\to c_2 \left (\frac {3 t^4}{320}+\frac {3 t^3}{40}+\frac {3 t^2}{8}+t+1\right )+c_1 \left (\frac {279 t^4+528 t^3+144 t^2-192 t+64}{64 t^2}-\frac {9}{16} \left (3 t^2+8 t+8\right ) \log (t)\right ) \]