Internal problem ID [1810]
Internal file name [OUTPUT/1811_Sunday_June_05_2022_02_33_12_AM_26790908/index.tex]
Book: Differential equations and their applications, 3rd ed., M. Braun
Section: Section 2.8.2, Regular singular points, the method of Frobenius. Page 214
Problem number: 18.
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "second order series method. Regular singular point. Difference is integer"
Maple gives the following as the ode type
[_Laguerre]
\[ \boxed {t y^{\prime \prime }-y^{\prime } \left (t +4\right )+2 y=0} \] With the expansion point for the power series method at \(t = 0\).
The type of the expansion point is first determined. This is done on the homogeneous part of the ODE. \[ t y^{\prime \prime }+\left (-t -4\right ) y^{\prime }+2 y = 0 \] The following is summary of singularities for the above ode. Writing the ode as \begin {align*} y^{\prime \prime }+p(t) y^{\prime } + q(t) y &=0 \end {align*}
Where \begin {align*} p(t) &= -\frac {t +4}{t}\\ q(t) &= \frac {2}{t}\\ \end {align*}
Combining everything together gives the following summary of singularities for the ode as
Regular singular points : \([0]\)
Irregular singular points : \([\infty ]\)
Since \(t = 0\) is regular singular point, then Frobenius power series is used. The ode is normalized to be \[ t y^{\prime \prime }+\left (-t -4\right ) y^{\prime }+2 y = 0 \] Let the solution be represented as Frobenius power series of the form \[ y = \moverset {\infty }{\munderset {n =0}{\sum }}a_{n} t^{n +r} \] Then \begin{align*} y^{\prime } &= \moverset {\infty }{\munderset {n =0}{\sum }}\left (n +r \right ) a_{n} t^{n +r -1} \\ y^{\prime \prime } &= \moverset {\infty }{\munderset {n =0}{\sum }}\left (n +r \right ) \left (n +r -1\right ) a_{n} t^{n +r -2} \\ \end{align*} Substituting the above back into the ode gives \begin{equation} \tag{1} t \left (\moverset {\infty }{\munderset {n =0}{\sum }}\left (n +r \right ) \left (n +r -1\right ) a_{n} t^{n +r -2}\right )+\left (-t -4\right ) \left (\moverset {\infty }{\munderset {n =0}{\sum }}\left (n +r \right ) a_{n} t^{n +r -1}\right )+2 \left (\moverset {\infty }{\munderset {n =0}{\sum }}a_{n} t^{n +r}\right ) = 0 \end{equation} Which simplifies to \begin{equation} \tag{2A} \left (\moverset {\infty }{\munderset {n =0}{\sum }}t^{n +r -1} a_{n} \left (n +r \right ) \left (n +r -1\right )\right )+\moverset {\infty }{\munderset {n =0}{\sum }}\left (-t^{n +r} a_{n} \left (n +r \right )\right )+\moverset {\infty }{\munderset {n =0}{\sum }}\left (-4 \left (n +r \right ) a_{n} t^{n +r -1}\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}2 a_{n} t^{n +r}\right ) = 0 \end{equation} The next step is to make all powers of \(t\) be \(n +r -1\) in each summation term. Going over each summation term above with power of \(t\) in it which is not already \(t^{n +r -1}\) and adjusting the power and the corresponding index gives \begin{align*} \moverset {\infty }{\munderset {n =0}{\sum }}\left (-t^{n +r} a_{n} \left (n +r \right )\right ) &= \moverset {\infty }{\munderset {n =1}{\sum }}\left (-a_{n -1} \left (n +r -1\right ) t^{n +r -1}\right ) \\ \moverset {\infty }{\munderset {n =0}{\sum }}2 a_{n} t^{n +r} &= \moverset {\infty }{\munderset {n =1}{\sum }}2 a_{n -1} t^{n +r -1} \\ \end{align*} Substituting all the above in Eq (2A) gives the following equation where now all powers of \(t\) are the same and equal to \(n +r -1\). \begin{equation} \tag{2B} \left (\moverset {\infty }{\munderset {n =0}{\sum }}t^{n +r -1} a_{n} \left (n +r \right ) \left (n +r -1\right )\right )+\moverset {\infty }{\munderset {n =1}{\sum }}\left (-a_{n -1} \left (n +r -1\right ) t^{n +r -1}\right )+\moverset {\infty }{\munderset {n =0}{\sum }}\left (-4 \left (n +r \right ) a_{n} t^{n +r -1}\right )+\left (\moverset {\infty }{\munderset {n =1}{\sum }}2 a_{n -1} t^{n +r -1}\right ) = 0 \end{equation} The indicial equation is obtained from \(n = 0\). From Eq (2B) this gives \[ t^{n +r -1} a_{n} \left (n +r \right ) \left (n +r -1\right )-4 \left (n +r \right ) a_{n} t^{n +r -1} = 0 \] When \(n = 0\) the above becomes \[ t^{-1+r} a_{0} r \left (-1+r \right )-4 r a_{0} t^{-1+r} = 0 \] Or \[ \left (t^{-1+r} r \left (-1+r \right )-4 r \,t^{-1+r}\right ) a_{0} = 0 \] Since \(a_{0}\neq 0\) then the above simplifies to \[ r \,t^{-1+r} \left (-5+r \right ) = 0 \] Since the above is true for all \(t\) then the indicial equation becomes \[ r \left (-5+r \right ) = 0 \] Solving for \(r\) gives the roots of the indicial equation as \begin {align*} r_1 &= 5\\ r_2 &= 0 \end {align*}
Since \(a_{0}\neq 0\) then the indicial equation becomes \[ r \,t^{-1+r} \left (-5+r \right ) = 0 \] Solving for \(r\) gives the roots of the indicial equation as \([5, 0]\).
Since \(r_1 - r_2 = 5\) is an integer, then we can construct two linearly independent solutions \begin {align*} y_{1}\left (t \right ) &= t^{r_{1}} \left (\moverset {\infty }{\munderset {n =0}{\sum }}a_{n} t^{n}\right )\\ y_{2}\left (t \right ) &= C y_{1}\left (t \right ) \ln \left (t \right )+t^{r_{2}} \left (\moverset {\infty }{\munderset {n =0}{\sum }}b_{n} t^{n}\right ) \end {align*}
Or \begin {align*} y_{1}\left (t \right ) &= t^{5} \left (\moverset {\infty }{\munderset {n =0}{\sum }}a_{n} t^{n}\right )\\ y_{2}\left (t \right ) &= C y_{1}\left (t \right ) \ln \left (t \right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}b_{n} t^{n}\right ) \end {align*}
Or \begin {align*} y_{1}\left (t \right ) &= \moverset {\infty }{\munderset {n =0}{\sum }}a_{n} t^{n +5}\\ y_{2}\left (t \right ) &= C y_{1}\left (t \right ) \ln \left (t \right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}b_{n} t^{n}\right ) \end {align*}
Where \(C\) above can be zero. We start by finding \(y_{1}\). Eq (2B) derived above is now used to find all \(a_{n}\) coefficients. The case \(n = 0\) is skipped since it was used to find the roots of the indicial equation. \(a_{0}\) is arbitrary and taken as \(a_{0} = 1\). For \(1\le n\) the recursive equation is \begin{equation} \tag{3} a_{n} \left (n +r \right ) \left (n +r -1\right )-a_{n -1} \left (n +r -1\right )-4 a_{n} \left (n +r \right )+2 a_{n -1} = 0 \end{equation} Solving for \(a_{n}\) from recursive equation (4) gives \[ a_{n} = \frac {a_{n -1} \left (n +r -3\right )}{n^{2}+2 n r +r^{2}-5 n -5 r}\tag {4} \] Which for the root \(r = 5\) becomes \[ a_{n} = \frac {a_{n -1} \left (n +2\right )}{n \left (n +5\right )}\tag {5} \] At this point, it is a good idea to keep track of \(a_{n}\) in a table both before substituting \(r = 5\) and after as more terms are found using the above recursive equation.
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
For \(n = 1\), using the above recursive equation gives \[ a_{1}=\frac {r -2}{r^{2}-3 r -4} \] Which for the root \(r = 5\) becomes \[ a_{1}={\frac {1}{2}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {r -2}{r^{2}-3 r -4}\) | \(\frac {1}{2}\) |
For \(n = 2\), using the above recursive equation gives \[ a_{2}=\frac {\left (-1+r \right ) \left (r -2\right )}{r^{4}-4 r^{3}-7 r^{2}+22 r +24} \] Which for the root \(r = 5\) becomes \[ a_{2}={\frac {1}{7}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {r -2}{r^{2}-3 r -4}\) | \(\frac {1}{2}\) |
| \(a_{2}\) | \(\frac {\left (-1+r \right ) \left (r -2\right )}{r^{4}-4 r^{3}-7 r^{2}+22 r +24}\) | \(\frac {1}{7}\) |
For \(n = 3\), using the above recursive equation gives \[ a_{3}=\frac {r \left (-1+r \right )}{r^{5}-r^{4}-19 r^{3}+r^{2}+90 r +72} \] Which for the root \(r = 5\) becomes \[ a_{3}={\frac {5}{168}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {r -2}{r^{2}-3 r -4}\) | \(\frac {1}{2}\) |
| \(a_{2}\) | \(\frac {\left (-1+r \right ) \left (r -2\right )}{r^{4}-4 r^{3}-7 r^{2}+22 r +24}\) | \(\frac {1}{7}\) |
| \(a_{3}\) | \(\frac {r \left (-1+r \right )}{r^{5}-r^{4}-19 r^{3}+r^{2}+90 r +72}\) | \(\frac {5}{168}\) |
For \(n = 4\), using the above recursive equation gives \[ a_{4}=\frac {r}{r^{5}+2 r^{4}-25 r^{3}-50 r^{2}+144 r +288} \] Which for the root \(r = 5\) becomes \[ a_{4}={\frac {5}{1008}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {r -2}{r^{2}-3 r -4}\) | \(\frac {1}{2}\) |
| \(a_{2}\) | \(\frac {\left (-1+r \right ) \left (r -2\right )}{r^{4}-4 r^{3}-7 r^{2}+22 r +24}\) | \(\frac {1}{7}\) |
| \(a_{3}\) | \(\frac {r \left (-1+r \right )}{r^{5}-r^{4}-19 r^{3}+r^{2}+90 r +72}\) | \(\frac {5}{168}\) |
| \(a_{4}\) | \(\frac {r}{r^{5}+2 r^{4}-25 r^{3}-50 r^{2}+144 r +288}\) | \(\frac {5}{1008}\) |
For \(n = 5\), using the above recursive equation gives \[ a_{5}=\frac {1}{\left (r +5\right ) \left (r^{4}-25 r^{2}+144\right )} \] Which for the root \(r = 5\) becomes \[ a_{5}={\frac {1}{1440}} \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {r -2}{r^{2}-3 r -4}\) | \(\frac {1}{2}\) |
| \(a_{2}\) | \(\frac {\left (-1+r \right ) \left (r -2\right )}{r^{4}-4 r^{3}-7 r^{2}+22 r +24}\) | \(\frac {1}{7}\) |
| \(a_{3}\) | \(\frac {r \left (-1+r \right )}{r^{5}-r^{4}-19 r^{3}+r^{2}+90 r +72}\) | \(\frac {5}{168}\) |
| \(a_{4}\) | \(\frac {r}{r^{5}+2 r^{4}-25 r^{3}-50 r^{2}+144 r +288}\) | \(\frac {5}{1008}\) |
| \(a_{5}\) | \(\frac {1}{\left (r +5\right ) \left (r^{4}-25 r^{2}+144\right )}\) | \(\frac {1}{1440}\) |
Using the above table, then the solution \(y_{1}\left (t \right )\) is \begin {align*} y_{1}\left (t \right )&= t^{5} \left (a_{0}+a_{1} t +a_{2} t^{2}+a_{3} t^{3}+a_{4} t^{4}+a_{5} t^{5}+a_{6} t^{6}\dots \right ) \\ &= t^{5} \left (1+\frac {t}{2}+\frac {t^{2}}{7}+\frac {5 t^{3}}{168}+\frac {5 t^{4}}{1008}+\frac {t^{5}}{1440}+O\left (t^{6}\right )\right ) \end {align*}
Now the second solution \(y_{2}\left (t \right )\) is found. Let \[ r_{1}-r_{2} = N \] Where \(N\) is positive integer which is the difference between the two roots. \(r_{1}\) is taken as the larger root. Hence for this problem we have \(N=5\). Now we need to determine if \(C\) is zero or not. This is done by finding \(\lim _{r\rightarrow r_{2}}a_{5}\left (r \right )\). If this limit exists, then \(C = 0\), else we need to keep the log term and \(C \neq 0\). The above table shows that \begin {align*} a_N &= a_{5} \\ &= \frac {1}{\left (r +5\right ) \left (r^{4}-25 r^{2}+144\right )} \end {align*}
Therefore \begin {align*} \lim _{r\rightarrow r_{2}}\frac {1}{\left (r +5\right ) \left (r^{4}-25 r^{2}+144\right )}&= \lim _{r\rightarrow 0}\frac {1}{\left (r +5\right ) \left (r^{4}-25 r^{2}+144\right )}\\ &= {\frac {1}{720}} \end {align*}
The limit is \(\frac {1}{720}\). Since the limit exists then the log term is not needed and we can set \(C = 0\). Therefore the second solution has the form \begin {align*} y_{2}\left (t \right ) &= \moverset {\infty }{\munderset {n =0}{\sum }}b_{n} t^{n +r}\\ &= \moverset {\infty }{\munderset {n =0}{\sum }}b_{n} t^{n} \end {align*}
Eq (3) derived above is used to find all \(b_{n}\) coefficients. The case \(n = 0\) is skipped since it was used to find the roots of the indicial equation. \(b_{0}\) is arbitrary and taken as \(b_{0} = 1\). For \(1\le n\) the recursive equation is \begin{equation} \tag{4} b_{n} \left (n +r \right ) \left (n +r -1\right )-b_{n -1} \left (n +r -1\right )-4 \left (n +r \right ) b_{n}+2 b_{n -1} = 0 \end{equation} Which for for the root \(r = 0\) becomes \begin{equation} \tag{4A} b_{n} n \left (n -1\right )-b_{n -1} \left (n -1\right )-4 n b_{n}+2 b_{n -1} = 0 \end{equation} Solving for \(b_{n}\) from the recursive equation (4) gives \[ b_{n} = \frac {b_{n -1} \left (n +r -3\right )}{n^{2}+2 n r +r^{2}-5 n -5 r}\tag {5} \] Which for the root \(r = 0\) becomes \[ b_{n} = \frac {b_{n -1} \left (n -3\right )}{n^{2}-5 n}\tag {6} \] At this point, it is a good idea to keep track of \(b_{n}\) in a table both before substituting \(r = 0\) and after as more terms are found using the above recursive equation.
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
For \(n = 1\), using the above recursive equation gives \[ b_{1}=\frac {r -2}{r^{2}-3 r -4} \] Which for the root \(r = 0\) becomes \[ b_{1}={\frac {1}{2}} \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(\frac {r -2}{r^{2}-3 r -4}\) | \(\frac {1}{2}\) |
For \(n = 2\), using the above recursive equation gives \[ b_{2}=\frac {\left (r -2\right ) \left (-1+r \right )}{\left (r^{2}-3 r -4\right ) \left (r^{2}-r -6\right )} \] Which for the root \(r = 0\) becomes \[ b_{2}={\frac {1}{12}} \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(\frac {r -2}{r^{2}-3 r -4}\) | \(\frac {1}{2}\) |
| \(b_{2}\) | \(\frac {\left (-1+r \right ) \left (r -2\right )}{r^{4}-4 r^{3}-7 r^{2}+22 r +24}\) | \(\frac {1}{12}\) |
For \(n = 3\), using the above recursive equation gives \[ b_{3}=\frac {r \left (-1+r \right )}{\left (r +3\right ) \left (r^{2}-3 r -4\right ) \left (r^{2}-r -6\right )} \] Which for the root \(r = 0\) becomes \[ b_{3}=0 \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(\frac {r -2}{r^{2}-3 r -4}\) | \(\frac {1}{2}\) |
| \(b_{2}\) | \(\frac {\left (-1+r \right ) \left (r -2\right )}{r^{4}-4 r^{3}-7 r^{2}+22 r +24}\) | \(\frac {1}{12}\) |
| \(b_{3}\) | \(\frac {r \left (-1+r \right )}{r^{5}-r^{4}-19 r^{3}+r^{2}+90 r +72}\) | \(0\) |
For \(n = 4\), using the above recursive equation gives \[ b_{4}=\frac {r}{\left (r +4\right ) \left (r^{2}-r -6\right ) \left (r -4\right ) \left (r +3\right )} \] Which for the root \(r = 0\) becomes \[ b_{4}=0 \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(\frac {r -2}{r^{2}-3 r -4}\) | \(\frac {1}{2}\) |
| \(b_{2}\) | \(\frac {\left (-1+r \right ) \left (r -2\right )}{r^{4}-4 r^{3}-7 r^{2}+22 r +24}\) | \(\frac {1}{12}\) |
| \(b_{3}\) | \(\frac {r \left (-1+r \right )}{r^{5}-r^{4}-19 r^{3}+r^{2}+90 r +72}\) | \(0\) |
| \(b_{4}\) | \(\frac {r}{\left (r^{2}-r -6\right ) \left (r +3\right ) \left (r^{2}-16\right )}\) | \(0\) |
For \(n = 5\), using the above recursive equation gives \[ b_{5}=\frac {1}{\left (r +3\right ) \left (r -4\right ) \left (r -3\right ) \left (r +4\right ) \left (r +5\right )} \] Which for the root \(r = 0\) becomes \[ b_{5}={\frac {1}{720}} \] And the table now becomes
| \(n\) | \(b_{n ,r}\) | \(b_{n}\) |
| \(b_{0}\) | \(1\) | \(1\) |
| \(b_{1}\) | \(\frac {r -2}{r^{2}-3 r -4}\) | \(\frac {1}{2}\) |
| \(b_{2}\) | \(\frac {\left (-1+r \right ) \left (r -2\right )}{r^{4}-4 r^{3}-7 r^{2}+22 r +24}\) | \(\frac {1}{12}\) |
| \(b_{3}\) | \(\frac {r \left (-1+r \right )}{r^{5}-r^{4}-19 r^{3}+r^{2}+90 r +72}\) | \(0\) |
| \(b_{4}\) | \(\frac {r}{\left (r^{2}-r -6\right ) \left (r +3\right ) \left (r^{2}-16\right )}\) | \(0\) |
| \(b_{5}\) | \(\frac {1}{\left (r +5\right ) \left (r^{2}-9\right ) \left (r^{2}-16\right )}\) | \(\frac {1}{720}\) |
Using the above table, then the solution \(y_{2}\left (t \right )\) is \begin {align*} y_{2}\left (t \right )&= b_{0}+b_{1} t +b_{2} t^{2}+b_{3} t^{3}+b_{4} t^{4}+b_{5} t^{5}+b_{6} t^{6}\dots \\ &= 1+\frac {t}{2}+\frac {t^{2}}{12}+\frac {t^{5}}{720}+O\left (t^{6}\right ) \end {align*}
Therefore the homogeneous solution is \begin{align*} y_h(t) &= c_{1} y_{1}\left (t \right )+c_{2} y_{2}\left (t \right ) \\ &= c_{1} t^{5} \left (1+\frac {t}{2}+\frac {t^{2}}{7}+\frac {5 t^{3}}{168}+\frac {5 t^{4}}{1008}+\frac {t^{5}}{1440}+O\left (t^{6}\right )\right ) + c_{2} \left (1+\frac {t}{2}+\frac {t^{2}}{12}+\frac {t^{5}}{720}+O\left (t^{6}\right )\right ) \\ \end{align*} Hence the final solution is \begin{align*} y &= y_h \\ &= c_{1} t^{5} \left (1+\frac {t}{2}+\frac {t^{2}}{7}+\frac {5 t^{3}}{168}+\frac {5 t^{4}}{1008}+\frac {t^{5}}{1440}+O\left (t^{6}\right )\right )+c_{2} \left (1+\frac {t}{2}+\frac {t^{2}}{12}+\frac {t^{5}}{720}+O\left (t^{6}\right )\right ) \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} t^{5} \left (1+\frac {t}{2}+\frac {t^{2}}{7}+\frac {5 t^{3}}{168}+\frac {5 t^{4}}{1008}+\frac {t^{5}}{1440}+O\left (t^{6}\right )\right )+c_{2} \left (1+\frac {t}{2}+\frac {t^{2}}{12}+\frac {t^{5}}{720}+O\left (t^{6}\right )\right ) \\ \end{align*}
Verification of solutions
\[ y = c_{1} t^{5} \left (1+\frac {t}{2}+\frac {t^{2}}{7}+\frac {5 t^{3}}{168}+\frac {5 t^{4}}{1008}+\frac {t^{5}}{1440}+O\left (t^{6}\right )\right )+c_{2} \left (1+\frac {t}{2}+\frac {t^{2}}{12}+\frac {t^{5}}{720}+O\left (t^{6}\right )\right ) \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & t y^{\prime \prime }+\left (-t -4\right ) y^{\prime }+2 y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 2 \\ {} & {} & y^{\prime \prime } \\ \bullet & {} & \textrm {Isolate 2nd derivative}\hspace {3pt} \\ {} & {} & y^{\prime \prime }=-\frac {2 y}{t}+\frac {\left (t +4\right ) y^{\prime }}{t} \\ \bullet & {} & \textrm {Group terms with}\hspace {3pt} y\hspace {3pt}\textrm {on the lhs of the ODE and the rest on the rhs of the ODE; ODE is linear}\hspace {3pt} \\ {} & {} & y^{\prime \prime }-\frac {\left (t +4\right ) y^{\prime }}{t}+\frac {2 y}{t}=0 \\ \square & {} & \textrm {Check to see if}\hspace {3pt} t_{0}=0\hspace {3pt}\textrm {is a regular singular point}\hspace {3pt} \\ {} & \circ & \textrm {Define functions}\hspace {3pt} \\ {} & {} & \left [P_{2}\left (t \right )=-\frac {t +4}{t}, P_{3}\left (t \right )=\frac {2}{t}\right ] \\ {} & \circ & t \cdot P_{2}\left (t \right )\textrm {is analytic at}\hspace {3pt} t =0 \\ {} & {} & \left (t \cdot P_{2}\left (t \right )\right )\bigg | {\mstack {}{_{t \hiderel {=}0}}}=-4 \\ {} & \circ & t^{2}\cdot P_{3}\left (t \right )\textrm {is analytic at}\hspace {3pt} t =0 \\ {} & {} & \left (t^{2}\cdot P_{3}\left (t \right )\right )\bigg | {\mstack {}{_{t \hiderel {=}0}}}=0 \\ {} & \circ & t =0\textrm {is a regular singular point}\hspace {3pt} \\ & {} & \textrm {Check to see if}\hspace {3pt} t_{0}=0\hspace {3pt}\textrm {is a regular singular point}\hspace {3pt} \\ {} & {} & t_{0}=0 \\ \bullet & {} & \textrm {Multiply by denominators}\hspace {3pt} \\ {} & {} & t y^{\prime \prime }+\left (-t -4\right ) y^{\prime }+2 y=0 \\ \bullet & {} & \textrm {Assume series solution for}\hspace {3pt} y \\ {} & {} & y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} t^{k +r} \\ \square & {} & \textrm {Rewrite ODE with series expansions}\hspace {3pt} \\ {} & \circ & \textrm {Convert}\hspace {3pt} t^{m}\cdot y^{\prime }\hspace {3pt}\textrm {to series expansion for}\hspace {3pt} m =0..1 \\ {} & {} & t^{m}\cdot y^{\prime }=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (k +r \right ) t^{k +r -1+m} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +1-m \\ {} & {} & t^{m}\cdot y^{\prime }=\moverset {\infty }{\munderset {k =-1+m}{\sum }}a_{k +1-m} \left (k +1-m +r \right ) t^{k +r} \\ {} & \circ & \textrm {Convert}\hspace {3pt} t \cdot y^{\prime \prime }\hspace {3pt}\textrm {to series expansion}\hspace {3pt} \\ {} & {} & t \cdot y^{\prime \prime }=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (k +r \right ) \left (k +r -1\right ) t^{k +r -1} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +1 \\ {} & {} & t \cdot y^{\prime \prime }=\moverset {\infty }{\munderset {k =-1}{\sum }}a_{k +1} \left (k +1+r \right ) \left (k +r \right ) t^{k +r} \\ & {} & \textrm {Rewrite ODE with series expansions}\hspace {3pt} \\ {} & {} & a_{0} r \left (-5+r \right ) t^{-1+r}+\left (\moverset {\infty }{\munderset {k =0}{\sum }}\left (a_{k +1} \left (k +1+r \right ) \left (k -4+r \right )-a_{k} \left (k +r -2\right )\right ) t^{k +r}\right )=0 \\ \bullet & {} & a_{0}\textrm {cannot be 0 by assumption, giving the indicial equation}\hspace {3pt} \\ {} & {} & r \left (-5+r \right )=0 \\ \bullet & {} & \textrm {Values of r that satisfy the indicial equation}\hspace {3pt} \\ {} & {} & r \in \left \{0, 5\right \} \\ \bullet & {} & \textrm {Each term in the series must be 0, giving the recursion relation}\hspace {3pt} \\ {} & {} & a_{k +1} \left (k +1+r \right ) \left (k -4+r \right )-a_{k} \left (k +r -2\right )=0 \\ \bullet & {} & \textrm {Recursion relation that defines series solution to ODE}\hspace {3pt} \\ {} & {} & a_{k +1}=\frac {a_{k} \left (k +r -2\right )}{\left (k +1+r \right ) \left (k -4+r \right )} \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =0\hspace {3pt}\textrm {; series terminates at}\hspace {3pt} k =2 \\ {} & {} & a_{k +1}=\frac {a_{k} \left (k -2\right )}{\left (k +1\right ) \left (k -4\right )} \\ \bullet & {} & \textrm {Apply recursion relation for}\hspace {3pt} k =0 \\ {} & {} & a_{1}=\frac {a_{0}}{2} \\ \bullet & {} & \textrm {Apply recursion relation for}\hspace {3pt} k =1 \\ {} & {} & a_{2}=\frac {a_{1}}{6} \\ \bullet & {} & \textrm {Express in terms of}\hspace {3pt} a_{0} \\ {} & {} & a_{2}=\frac {a_{0}}{12} \\ \bullet & {} & \textrm {Terminating series solution of the ODE for}\hspace {3pt} r =0\hspace {3pt}\textrm {. Use reduction of order to find the second linearly independent solution}\hspace {3pt} \\ {} & {} & y=a_{0}\cdot \left (\frac {1}{2} t +1+\frac {1}{12} t^{2}\right ) \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =5 \\ {} & {} & a_{k +1}=\frac {a_{k} \left (k +3\right )}{\left (k +6\right ) \left (k +1\right )} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =5 \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} t^{k +5}, a_{k +1}=\frac {a_{k} \left (k +3\right )}{\left (k +6\right ) \left (k +1\right )}\right ] \\ \bullet & {} & \textrm {Combine solutions and rename parameters}\hspace {3pt} \\ {} & {} & \left [y=a_{0}\cdot \left (\frac {1}{2} t +1+\frac {1}{12} t^{2}\right )+\left (\moverset {\infty }{\munderset {k =0}{\sum }}b_{k} t^{5+k}\right ), b_{1+k}=\frac {b_{k} \left (k +3\right )}{\left (k +6\right ) \left (1+k \right )}\right ] \end {array} \]
Maple trace Kovacic algorithm successful
`Methods for second order ODEs: --- Trying classification methods --- trying a quadrature checking if the LODE has constant coefficients checking if the LODE is of Euler type trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Trying a Liouvillian solution using Kovacics algorithm A Liouvillian solution exists Reducible group (found an exponential solution) Reducible group (found another exponential solution) <- Kovacics algorithm successful`
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 40
Order:=6; dsolve(t*diff(y(t),t$2)-(4+t)*diff(y(t),t)+2*y(t)=0,y(t),type='series',t=0);
\[ y \left (t \right ) = c_{1} t^{5} \left (1+\frac {1}{2} t +\frac {1}{7} t^{2}+\frac {5}{168} t^{3}+\frac {5}{1008} t^{4}+\frac {1}{1440} t^{5}+\operatorname {O}\left (t^{6}\right )\right )+c_{2} \left (2880+1440 t +240 t^{2}+4 t^{5}+\operatorname {O}\left (t^{6}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.034 (sec). Leaf size: 56
AsymptoticDSolveValue[t*y''[t]-(4+t)*y'[t]+2*y[t]==0,y[t],{t,0,5}]
\[ y(t)\to c_1 \left (\frac {t^2}{12}+\frac {t}{2}+1\right )+c_2 \left (\frac {5 t^9}{1008}+\frac {5 t^8}{168}+\frac {t^7}{7}+\frac {t^6}{2}+t^5\right ) \]