14.14.11 problem 11

Internal problem ID [2648]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.8.2, Regular singular points, the method of Frobenius. Excercises page 216
Problem number : 11
Date solved : Monday, January 27, 2025 at 06:05:20 AM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} 4 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: 0.010 (sec). Leaf size: 44

Order:=6; 
dsolve(4*t*diff(y(t),t$2)+3*diff(y(t),t)-3*y(t)=0,y(t),type='series',t=0);
 

\[ y = c_1 \,t^{{1}/{4}} \left (1+\frac {3}{5} t +\frac {1}{10} t^{2}+\frac {1}{130} t^{3}+\frac {3}{8840} t^{4}+\frac {3}{309400} t^{5}+\operatorname {O}\left (t^{6}\right )\right )+c_2 \left (1+t +\frac {3}{14} t^{2}+\frac {3}{154} t^{3}+\frac {3}{3080} t^{4}+\frac {9}{292600} t^{5}+\operatorname {O}\left (t^{6}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 81

AsymptoticDSolveValue[4*t*D[y[t],{t,2}]+3*D[y[t],t]-3*y[t]==0,y[t],{t,0,"6"-1}]
 

\[ y(t)\to c_1 \sqrt [4]{t} \left (\frac {3 t^5}{309400}+\frac {3 t^4}{8840}+\frac {t^3}{130}+\frac {t^2}{10}+\frac {3 t}{5}+1\right )+c_2 \left (\frac {9 t^5}{292600}+\frac {3 t^4}{3080}+\frac {3 t^3}{154}+\frac {3 t^2}{14}+t+1\right ) \]