7.15.41 problem 42
Internal
problem
ID
[497]
Book
:
Elementary
Differential
Equations.
By
C.
Henry
Edwards,
David
E.
Penney
and
David
Calvis.
6th
edition.
2008
Section
:
Chapter
3.
Power
series
methods.
Section
3.3
(Regular
singular
points).
Problems
at
page
231
Problem
number
:
42
Date
solved
:
Monday, January 27, 2025 at 02:54:09 AM
CAS
classification
:
[_Jacobi]
\begin{align*} x \left (1-x \right ) y^{\prime \prime }+\left (\gamma -\left (\alpha +\beta +1\right ) x \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: 0.018 (sec). Leaf size: 447
Order:=6;
dsolve(x*(1-x)*diff(y(x),x$2)+(gamma-(alpha+beta+1)*x)*diff(y(x),x)-alpha*beta*y(x)=0,y(x),type='series',x=0);
\[
y = c_1 \,x^{-\gamma +1} \left (1-\frac {\left (\alpha -\gamma +1\right ) \left (\beta -\gamma +1\right )}{\gamma -2} x +\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 )} x^{2}-\frac {1}{6} \frac {\left (\alpha -\gamma +2\right ) \left (\alpha -\gamma +1\right ) \left (-\gamma +3+\alpha \right ) \left (\beta -\gamma +1\right ) \left (-\gamma +3+\beta \right ) \left (\beta -\gamma +2\right )}{\left (\gamma -2\right ) \left (\gamma -3\right ) \left (\gamma -4\right )} x^{3}+\frac {1}{24} \frac {\left (\alpha -\gamma +1\right ) \left (-\gamma +4+\alpha \right ) \left (-\gamma +3+\alpha \right ) \left (\alpha -\gamma +2\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 )} x^{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 (\beta -\gamma +2\right ) \left (\beta -\gamma +1\right ) \left (-\gamma +5+\beta \right ) \left (-\gamma +4+\beta \right ) \left (-\gamma +3+\beta \right )}{\left (\gamma -2\right ) \left (\gamma -3\right ) \left (\gamma -4\right ) \left (\gamma -5\right ) \left (\gamma -6\right )} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (1+\frac {\alpha \beta }{\gamma } x +\frac {1}{2} \frac {\alpha \beta \left (1+\alpha \right ) \left (\beta +1\right )}{\gamma \left (\gamma +1\right )} x^{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 )} x^{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 )} x^{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 )} x^{5}+\operatorname {O}\left (x^{6}\right )\right )
\]
✓ Solution by Mathematica
Time used: 0.007 (sec). Leaf size: 2568
AsymptoticDSolveValue[x*(1-x)*D[y[x],{x,2}]+(\[Gamma]-(\[Alpha]+\[Beta]+1)*x)*D[y[x],x]-\[Alpha]*\[Beta]*y[x]==0,y[x],{x,0,"6"-1}]
Too large to display