14.14.26 problem 26
Internal
problem
ID
[2663]
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
:
26
Date
solved
:
Monday, January 27, 2025 at 06:05:39 AM
CAS
classification
:
[_Jacobi]
\begin{align*} t \left (1-t \right ) y^{\prime \prime }+\left (\gamma -\left (\alpha +\beta +1\right ) t \right ) y^{\prime }-\alpha \beta y&=0 \end{align*}
Using series method with expansion around
\begin{align*} 0 \end{align*}
✓ Solution by Maple
Time used: 0.023 (sec). Leaf size: 447
Order:=6;
dsolve(t*(1-t)*diff(y(t),t$2)+(gamma-(1+alpha+beta)*t)*diff(y(t),t)-alpha*beta*y(t)=0,y(t),type='series',t=0);
\[
y = c_1 \,t^{-\gamma +1} \left (1-\frac {\left (\alpha -\gamma +1\right ) \left (\beta -\gamma +1\right )}{\gamma -2} t +\frac {1}{2} \frac {\left (\alpha -\gamma +2\right ) \left (\alpha -\gamma +1\right ) \left (\beta -\gamma +2\right ) \left (\beta -\gamma +1\right )}{\left (\gamma -2\right ) \left (\gamma -3\right )} t^{2}-\frac {1}{6} \frac {\left (\alpha -\gamma +2\right ) \left (\alpha -\gamma +1\right ) \left (-\gamma +3+\alpha \right ) \left (\beta -\gamma +2\right ) \left (\beta -\gamma +1\right ) \left (-\gamma +3+\beta \right )}{\left (\gamma -2\right ) \left (\gamma -3\right ) \left (\gamma -4\right )} t^{3}+\frac {1}{24} \frac {\left (-\gamma +3+\alpha \right ) \left (\alpha -\gamma +2\right ) \left (\alpha -\gamma +1\right ) \left (-\gamma +4+\alpha \right ) \left (\beta -\gamma +1\right ) \left (-\gamma +4+\beta \right ) \left (-\gamma +3+\beta \right ) \left (\beta -\gamma +2\right )}{\left (\gamma -2\right ) \left (\gamma -3\right ) \left (\gamma -4\right ) \left (\gamma -5\right )} t^{4}-\frac {1}{120} \frac {\left (-\gamma +4+\alpha \right ) \left (-\gamma +3+\alpha \right ) \left (\alpha -\gamma +2\right ) \left (\alpha -\gamma +1\right ) \left (-\gamma +5+\alpha \right ) \left (-\gamma +3+\beta \right ) \left (\beta -\gamma +2\right ) \left (\beta -\gamma +1\right ) \left (-\gamma +5+\beta \right ) \left (-\gamma +4+\beta \right )}{\left (\gamma -2\right ) \left (\gamma -3\right ) \left (\gamma -4\right ) \left (\gamma -5\right ) \left (\gamma -6\right )} t^{5}+\operatorname {O}\left (t^{6}\right )\right )+c_2 \left (1+\frac {\alpha \beta }{\gamma } t +\frac {1}{2} \frac {\alpha \beta \left (1+\alpha \right ) \left (\beta +1\right )}{\gamma \left (\gamma +1\right )} t^{2}+\frac {1}{6} \frac {\alpha \beta \left (\alpha +2\right ) \left (1+\alpha \right ) \left (\beta +2\right ) \left (\beta +1\right )}{\gamma \left (\gamma +1\right ) \left (\gamma +2\right )} t^{3}+\frac {1}{24} \frac {\alpha \beta \left (\alpha +3\right ) \left (\alpha +2\right ) \left (1+\alpha \right ) \left (\beta +3\right ) \left (\beta +2\right ) \left (\beta +1\right )}{\gamma \left (\gamma +1\right ) \left (\gamma +2\right ) \left (\gamma +3\right )} t^{4}+\frac {1}{120} \frac {\alpha \beta \left (\alpha +4\right ) \left (\alpha +3\right ) \left (\alpha +2\right ) \left (1+\alpha \right ) \left (\beta +4\right ) \left (\beta +3\right ) \left (\beta +2\right ) \left (\beta +1\right )}{\gamma \left (\gamma +1\right ) \left (\gamma +2\right ) \left (\gamma +3\right ) \left (\gamma +4\right )} t^{5}+\operatorname {O}\left (t^{6}\right )\right )
\]
✓ Solution by Mathematica
Time used: 0.009 (sec). Leaf size: 2342
AsymptoticDSolveValue[t*(1-t)*D[y[t],{t,2}]+(\[Gamma]-(1+\[Alpha]*\[Beta])*t)*D[y[t],t]-\[Alpha]*\[Beta]*y[t]==0,y[t],{t,0,"6"-1}]
Too large to display