14.14.28 problem 29

Internal problem ID [2665]
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 : 29
Date solved : Monday, January 27, 2025 at 06:05:42 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: 0.008 (sec). Leaf size: 47

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);
 

\[ y = c_1 \,t^{-i} \left (1+\left (-\frac {1}{5}-\frac {2 i}{5}\right ) t +\left (-\frac {1}{40}+\frac {3 i}{40}\right ) t^{2}+\left (\frac {3}{520}-\frac {7 i}{1560}\right ) t^{3}+\left (-\frac {1}{2496}+\frac {i}{12480}\right ) t^{4}+\left (\frac {9}{603200}+\frac {i}{361920}\right ) t^{5}+\operatorname {O}\left (t^{6}\right )\right )+c_2 \,t^{i} \left (1+\left (-\frac {1}{5}+\frac {2 i}{5}\right ) t +\left (-\frac {1}{40}-\frac {3 i}{40}\right ) t^{2}+\left (\frac {3}{520}+\frac {7 i}{1560}\right ) t^{3}+\left (-\frac {1}{2496}-\frac {i}{12480}\right ) t^{4}+\left (\frac {9}{603200}-\frac {i}{361920}\right ) t^{5}+\operatorname {O}\left (t^{6}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.012 (sec). Leaf size: 90

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

\[ y(t)\to \left (\frac {1}{12480}+\frac {i}{2496}\right ) c_2 t^{-i} \left (i t^4-(8+16 i) t^3+(168+96 i) t^2-(1056-288 i) t+(480-2400 i)\right )-\left (\frac {1}{2496}+\frac {i}{12480}\right ) c_1 t^i \left (t^4-(16+8 i) t^3+(96+168 i) t^2+(288-1056 i) t-(2400-480 i)\right ) \]