Internal problem ID [1400]
Internal file name [OUTPUT/1401_Sunday_June_05_2022_02_15_14_AM_70027925/index.tex]
Book: Elementary differential equations with boundary value problems. William F. Trench.
Brooks/Cole 2001
Section: Chapter 7 Series Solutions of Linear Second Equations. 7.6 THE METHOD OF
FROBENIUS II. Exercises 7.6. Page 374
Problem number: 48.
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "second order series method. Regular singular point. Repeated root"
Maple gives the following as the ode type
[[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{2} \left (1-x \right ) y^{\prime \prime }-x \left (3-5 x \right ) y^{\prime }+\left (4-5 x \right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
The type of the expansion point is first determined. This is done on the homogeneous part of the ODE. \[ \left (-x^{3}+x^{2}\right ) y^{\prime \prime }+\left (5 x^{2}-3 x \right ) y^{\prime }+\left (4-5 x \right ) y = 0 \] The following is summary of singularities for the above ode. Writing the ode as \begin {align*} y^{\prime \prime }+p(x) y^{\prime } + q(x) y &=0 \end {align*}
Where \begin {align*} p(x) &= -\frac {5 x -3}{\left (-1+x \right ) x}\\ q(x) &= \frac {-4+5 x}{x^{2} \left (-1+x \right )}\\ \end {align*}
| \(p(x)=-\frac {5 x -3}{\left (-1+x \right ) x}\) | |
| singularity | type |
| \(x = 0\) | \(\text {``regular''}\) |
| \(x = 1\) | \(\text {``regular''}\) |
| \(q(x)=\frac {-4+5 x}{x^{2} \left (-1+x \right )}\) | |
| singularity | type |
| \(x = 0\) | \(\text {``regular''}\) |
| \(x = 1\) | \(\text {``regular''}\) |
Combining everything together gives the following summary of singularities for the ode as
Regular singular points : \([0, 1, \infty ]\)
Irregular singular points : \([]\)
Since \(x = 0\) is regular singular point, then Frobenius power series is used. The ode is normalized to be \[ -y^{\prime \prime } x^{2} \left (-1+x \right )+\left (5 x^{2}-3 x \right ) y^{\prime }+\left (4-5 x \right ) y = 0 \] Let the solution be represented as Frobenius power series of the form \[ y = \moverset {\infty }{\munderset {n =0}{\sum }}a_{n} x^{n +r} \] Then \begin{align*} y^{\prime } &= \moverset {\infty }{\munderset {n =0}{\sum }}\left (n +r \right ) a_{n} x^{n +r -1} \\ y^{\prime \prime } &= \moverset {\infty }{\munderset {n =0}{\sum }}\left (n +r \right ) \left (n +r -1\right ) a_{n} x^{n +r -2} \\ \end{align*} Substituting the above back into the ode gives \begin{equation} \tag{1} -\left (\moverset {\infty }{\munderset {n =0}{\sum }}\left (n +r \right ) \left (n +r -1\right ) a_{n} x^{n +r -2}\right ) x^{2} \left (-1+x \right )+\left (5 x^{2}-3 x \right ) \left (\moverset {\infty }{\munderset {n =0}{\sum }}\left (n +r \right ) a_{n} x^{n +r -1}\right )+\left (4-5 x \right ) \left (\moverset {\infty }{\munderset {n =0}{\sum }}a_{n} x^{n +r}\right ) = 0 \end{equation} Which simplifies to \begin{equation} \tag{2A} \moverset {\infty }{\munderset {n =0}{\sum }}\left (-x^{1+n +r} a_{n} \left (n +r \right ) \left (n +r -1\right )\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}x^{n +r} a_{n} \left (n +r \right ) \left (n +r -1\right )\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}5 x^{1+n +r} a_{n} \left (n +r \right )\right )+\moverset {\infty }{\munderset {n =0}{\sum }}\left (-3 x^{n +r} a_{n} \left (n +r \right )\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}4 a_{n} x^{n +r}\right )+\moverset {\infty }{\munderset {n =0}{\sum }}\left (-5 x^{1+n +r} a_{n}\right ) = 0 \end{equation} The next step is to make all powers of \(x\) be \(n +r\) in each summation term. Going over each summation term above with power of \(x\) in it which is not already \(x^{n +r}\) and adjusting the power and the corresponding index gives \begin{align*} \moverset {\infty }{\munderset {n =0}{\sum }}\left (-x^{1+n +r} 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 ) \left (n +r -2\right ) x^{n +r}\right ) \\ \moverset {\infty }{\munderset {n =0}{\sum }}5 x^{1+n +r} a_{n} \left (n +r \right ) &= \moverset {\infty }{\munderset {n =1}{\sum }}5 a_{n -1} \left (n +r -1\right ) x^{n +r} \\ \moverset {\infty }{\munderset {n =0}{\sum }}\left (-5 x^{1+n +r} a_{n}\right ) &= \moverset {\infty }{\munderset {n =1}{\sum }}\left (-5 a_{n -1} x^{n +r}\right ) \\ \end{align*} Substituting all the above in Eq (2A) gives the following equation where now all powers of \(x\) are the same and equal to \(n +r\). \begin{equation} \tag{2B} \moverset {\infty }{\munderset {n =1}{\sum }}\left (-a_{n -1} \left (n +r -1\right ) \left (n +r -2\right ) x^{n +r}\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}x^{n +r} a_{n} \left (n +r \right ) \left (n +r -1\right )\right )+\left (\moverset {\infty }{\munderset {n =1}{\sum }}5 a_{n -1} \left (n +r -1\right ) x^{n +r}\right )+\moverset {\infty }{\munderset {n =0}{\sum }}\left (-3 x^{n +r} a_{n} \left (n +r \right )\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}4 a_{n} x^{n +r}\right )+\moverset {\infty }{\munderset {n =1}{\sum }}\left (-5 a_{n -1} x^{n +r}\right ) = 0 \end{equation} The indicial equation is obtained from \(n = 0\). From Eq (2B) this gives \[ x^{n +r} a_{n} \left (n +r \right ) \left (n +r -1\right )-3 x^{n +r} a_{n} \left (n +r \right )+4 a_{n} x^{n +r} = 0 \] When \(n = 0\) the above becomes \[ x^{r} a_{0} r \left (-1+r \right )-3 x^{r} a_{0} r +4 a_{0} x^{r} = 0 \] Or \[ \left (x^{r} r \left (-1+r \right )-3 x^{r} r +4 x^{r}\right ) a_{0} = 0 \] Since \(a_{0}\neq 0\) then the above simplifies to \[ \left (r -2\right )^{2} x^{r} = 0 \] Since the above is true for all \(x\) then the indicial equation becomes \[ \left (r -2\right )^{2} = 0 \] Solving for \(r\) gives the roots of the indicial equation as \begin {align*} r_1 &= 2\\ r_2 &= 2 \end {align*}
Since \(a_{0}\neq 0\) then the indicial equation becomes \[ \left (r -2\right )^{2} x^{r} = 0 \] Solving for \(r\) gives the roots of the indicial equation as \([2, 2]\).
Since the root of the indicial equation is repeated, then we can construct two linearly independent solutions. The first solution has the form \begin {align*} y_{1}\left (x \right ) &= \moverset {\infty }{\munderset {n =0}{\sum }}a_{n} x^{n +r}\tag {1A} \end {align*}
Now the second solution \(y_{2}\) is found using \begin {align*} y_{2}\left (x \right ) &= y_{1}\left (x \right ) \ln \left (x \right )+\left (\moverset {\infty }{\munderset {n =1}{\sum }}b_{n} x^{n +r}\right )\tag {1B} \end {align*}
Then the general solution will be \[ y = c_{1} y_{1}\left (x \right )+c_{2} y_{2}\left (x \right ) \] In Eq (1B) the sum starts from 1 and not zero. In Eq (1A), \(a_{0}\) is never zero, and is arbitrary and is typically taken as \(a_{0} = 1\), and \(\{c_{1}, c_{2}\}\) are two arbitray constants of integration which can be found from initial conditions. Using the value of the indicial root found earlier, \(r = 2\), Eqs (1A,1B) become \begin {align*} y_{1}\left (x \right ) &= \moverset {\infty }{\munderset {n =0}{\sum }}a_{n} x^{n +2}\\ y_{2}\left (x \right ) &= y_{1}\left (x \right ) \ln \left (x \right )+\left (\moverset {\infty }{\munderset {n =1}{\sum }}b_{n} x^{n +2}\right ) \end {align*}
We start by finding the first solution \(y_{1}\left (x \right )\). 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 -1} \left (n +r -1\right ) \left (n +r -2\right )+a_{n} \left (n +r \right ) \left (n +r -1\right )+5 a_{n -1} \left (n +r -1\right )-3 a_{n} \left (n +r \right )+4 a_{n}-5 a_{n -1} = 0 \end{equation} Solving for \(a_{n}\) from recursive equation (4) gives \[ a_{n} = \frac {\left (n +r -6\right ) a_{n -1}}{n +r -2}\tag {4} \] Which for the root \(r = 2\) becomes \[ a_{n} = \frac {\left (n -4\right ) a_{n -1}}{n}\tag {5} \] At this point, it is a good idea to keep track of \(a_{n}\) in a table both before substituting \(r = 2\) 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 \[ a_{1}=\frac {-5+r}{-1+r} \] Which for the root \(r = 2\) becomes \[ a_{1}=-3 \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {-5+r}{-1+r}\) | \(-3\) |
For \(n = 2\), using the above recursive equation gives \[ a_{2}=\frac {r^{2}-9 r +20}{r \left (-1+r \right )} \] Which for the root \(r = 2\) becomes \[ a_{2}=3 \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {-5+r}{-1+r}\) | \(-3\) |
| \(a_{2}\) | \(\frac {r^{2}-9 r +20}{r \left (-1+r \right )}\) | \(3\) |
For \(n = 3\), using the above recursive equation gives \[ a_{3}=\frac {r^{3}-12 r^{2}+47 r -60}{r^{3}-r} \] Which for the root \(r = 2\) becomes \[ a_{3}=-1 \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {-5+r}{-1+r}\) | \(-3\) |
| \(a_{2}\) | \(\frac {r^{2}-9 r +20}{r \left (-1+r \right )}\) | \(3\) |
| \(a_{3}\) | \(\frac {r^{3}-12 r^{2}+47 r -60}{r^{3}-r}\) | \(-1\) |
For \(n = 4\), using the above recursive equation gives \[ a_{4}=\frac {r^{4}-14 r^{3}+71 r^{2}-154 r +120}{r \left (r^{2}-1\right ) \left (2+r \right )} \] Which for the root \(r = 2\) becomes \[ a_{4}=0 \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {-5+r}{-1+r}\) | \(-3\) |
| \(a_{2}\) | \(\frac {r^{2}-9 r +20}{r \left (-1+r \right )}\) | \(3\) |
| \(a_{3}\) | \(\frac {r^{3}-12 r^{2}+47 r -60}{r^{3}-r}\) | \(-1\) |
| \(a_{4}\) | \(\frac {r^{4}-14 r^{3}+71 r^{2}-154 r +120}{r \left (r^{2}-1\right ) \left (2+r \right )}\) | \(0\) |
For \(n = 5\), using the above recursive equation gives \[ a_{5}=\frac {r^{4}-14 r^{3}+71 r^{2}-154 r +120}{\left (3+r \right ) \left (2+r \right ) \left (1+r \right ) r} \] Which for the root \(r = 2\) becomes \[ a_{5}=0 \] And the table now becomes
| \(n\) | \(a_{n ,r}\) | \(a_{n}\) |
| \(a_{0}\) | \(1\) | \(1\) |
| \(a_{1}\) | \(\frac {-5+r}{-1+r}\) | \(-3\) |
| \(a_{2}\) | \(\frac {r^{2}-9 r +20}{r \left (-1+r \right )}\) | \(3\) |
| \(a_{3}\) | \(\frac {r^{3}-12 r^{2}+47 r -60}{r^{3}-r}\) | \(-1\) |
| \(a_{4}\) | \(\frac {r^{4}-14 r^{3}+71 r^{2}-154 r +120}{r \left (r^{2}-1\right ) \left (2+r \right )}\) | \(0\) |
| \(a_{5}\) | \(\frac {r^{4}-14 r^{3}+71 r^{2}-154 r +120}{\left (3+r \right ) \left (2+r \right ) \left (1+r \right ) r}\) | \(0\) |
Using the above table, then the first solution \(y_{1}\left (x \right )\) is \begin{align*} y_{1}\left (x \right )&= x^{2} \left (a_{0}+a_{1} x +a_{2} x^{2}+a_{3} x^{3}+a_{4} x^{4}+a_{5} x^{5}+a_{6} x^{6}\dots \right ) \\ &= x^{2} \left (-x^{3}+3 x^{2}-3 x +1+O\left (x^{6}\right )\right ) \\ \end{align*} Now the second solution is found. The second solution is given by \[ y_{2}\left (x \right ) = y_{1}\left (x \right ) \ln \left (x \right )+\left (\moverset {\infty }{\munderset {n =1}{\sum }}b_{n} x^{n +r}\right ) \] Where \(b_{n}\) is found using \[ b_{n} = \frac {d}{d r}a_{n ,r} \] And the above is then evaluated at \(r = 2\). The above table for \(a_{n ,r}\) is used for this purpose. Computing the derivatives gives the following table
| \(n\) | \(b_{n ,r}\) | \(a_{n}\) | \(b_{n ,r} = \frac {d}{d r}a_{n ,r}\) | \(b_{n}\left (r =2\right )\) |
| \(b_{0}\) | \(1\) | \(1\) | N/A since \(b_{n}\) starts from 1 | N/A |
| \(b_{1}\) | \(\frac {-5+r}{-1+r}\) | \(-3\) | \(\frac {4}{\left (-1+r \right )^{2}}\) | \(4\) |
| \(b_{2}\) | \(\frac {r^{2}-9 r +20}{r \left (-1+r \right )}\) | \(3\) | \(\frac {8 r^{2}-40 r +20}{r^{2} \left (-1+r \right )^{2}}\) | \(-7\) |
| \(b_{3}\) | \(\frac {r^{3}-12 r^{2}+47 r -60}{r^{3}-r}\) | \(-1\) | \(\frac {12 r^{4}-96 r^{3}+192 r^{2}-60}{r^{2} \left (r^{2}-1\right )^{2}}\) | \(\frac {11}{3}\) |
| \(b_{4}\) | \(\frac {r^{4}-14 r^{3}+71 r^{2}-154 r +120}{r \left (r^{2}-1\right ) \left (2+r \right )}\) | \(0\) | \(\frac {16 r^{6}-144 r^{5}+328 r^{4}+192 r^{3}-1016 r^{2}+240 r +240}{r^{2} \left (r^{2}-1\right )^{2} \left (2+r \right )^{2}}\) | \(-{\frac {1}{4}}\) |
| \(b_{5}\) | \(\frac {r^{4}-14 r^{3}+71 r^{2}-154 r +120}{\left (3+r \right ) \left (2+r \right ) \left (1+r \right ) r}\) | \(0\) | \(\frac {20 r^{6}-120 r^{5}-100 r^{4}+1200 r^{3}-40 r^{2}-2640 r -720}{\left (3+r \right )^{2} \left (2+r \right )^{2} \left (1+r \right )^{2} r^{2}}\) | \(-{\frac {1}{20}}\) |
The above table gives all values of \(b_{n}\) needed. Hence the second solution is \begin{align*} y_{2}\left (x \right )&=y_{1}\left (x \right ) \ln \left (x \right )+b_{0}+b_{1} x +b_{2} x^{2}+b_{3} x^{3}+b_{4} x^{4}+b_{5} x^{5}+b_{6} x^{6}\dots \\ &= x^{2} \left (-x^{3}+3 x^{2}-3 x +1+O\left (x^{6}\right )\right ) \ln \left (x \right )+x^{2} \left (-7 x^{2}+4 x +\frac {11 x^{3}}{3}-\frac {x^{4}}{4}-\frac {x^{5}}{20}+O\left (x^{6}\right )\right ) \\ \end{align*} Therefore the homogeneous solution is \begin{align*} y_h(x) &= c_{1} y_{1}\left (x \right )+c_{2} y_{2}\left (x \right ) \\ &= c_{1} x^{2} \left (-x^{3}+3 x^{2}-3 x +1+O\left (x^{6}\right )\right ) + c_{2} \left (x^{2} \left (-x^{3}+3 x^{2}-3 x +1+O\left (x^{6}\right )\right ) \ln \left (x \right )+x^{2} \left (-7 x^{2}+4 x +\frac {11 x^{3}}{3}-\frac {x^{4}}{4}-\frac {x^{5}}{20}+O\left (x^{6}\right )\right )\right ) \\ \end{align*} Hence the final solution is \begin{align*} y &= y_h \\ &= c_{1} x^{2} \left (-x^{3}+3 x^{2}-3 x +1+O\left (x^{6}\right )\right )+c_{2} \left (x^{2} \left (-x^{3}+3 x^{2}-3 x +1+O\left (x^{6}\right )\right ) \ln \left (x \right )+x^{2} \left (-7 x^{2}+4 x +\frac {11 x^{3}}{3}-\frac {x^{4}}{4}-\frac {x^{5}}{20}+O\left (x^{6}\right )\right )\right ) \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} x^{2} \left (-x^{3}+3 x^{2}-3 x +1+O\left (x^{6}\right )\right )+c_{2} \left (x^{2} \left (-x^{3}+3 x^{2}-3 x +1+O\left (x^{6}\right )\right ) \ln \left (x \right )+x^{2} \left (-7 x^{2}+4 x +\frac {11 x^{3}}{3}-\frac {x^{4}}{4}-\frac {x^{5}}{20}+O\left (x^{6}\right )\right )\right ) \\ \end{align*}
Verification of solutions
\[ y = c_{1} x^{2} \left (-x^{3}+3 x^{2}-3 x +1+O\left (x^{6}\right )\right )+c_{2} \left (x^{2} \left (-x^{3}+3 x^{2}-3 x +1+O\left (x^{6}\right )\right ) \ln \left (x \right )+x^{2} \left (-7 x^{2}+4 x +\frac {11 x^{3}}{3}-\frac {x^{4}}{4}-\frac {x^{5}}{20}+O\left (x^{6}\right )\right )\right ) \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & -\left (\frac {d}{d x}y^{\prime }\right ) x^{2} \left (-1+x \right )+\left (5 x^{2}-3 x \right ) y^{\prime }+\left (4-5 x \right ) y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 2 \\ {} & {} & \frac {d}{d x}y^{\prime } \\ \bullet & {} & \textrm {Isolate 2nd derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y^{\prime }=-\frac {\left (-4+5 x \right ) y}{x^{2} \left (-1+x \right )}+\frac {\left (5 x -3\right ) y^{\prime }}{x \left (-1+x \right )} \\ \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} \\ {} & {} & \frac {d}{d x}y^{\prime }-\frac {\left (5 x -3\right ) y^{\prime }}{x \left (-1+x \right )}+\frac {\left (-4+5 x \right ) y}{x^{2} \left (-1+x \right )}=0 \\ \square & {} & \textrm {Check to see if}\hspace {3pt} x_{0}\hspace {3pt}\textrm {is a regular singular point}\hspace {3pt} \\ {} & \circ & \textrm {Define functions}\hspace {3pt} \\ {} & {} & \left [P_{2}\left (x \right )=-\frac {5 x -3}{\left (-1+x \right ) x}, P_{3}\left (x \right )=\frac {-4+5 x}{x^{2} \left (-1+x \right )}\right ] \\ {} & \circ & x \cdot P_{2}\left (x \right )\textrm {is analytic at}\hspace {3pt} x =0 \\ {} & {} & \left (x \cdot P_{2}\left (x \right )\right )\bigg | {\mstack {}{_{x \hiderel {=}0}}}=-3 \\ {} & \circ & x^{2}\cdot P_{3}\left (x \right )\textrm {is analytic at}\hspace {3pt} x =0 \\ {} & {} & \left (x^{2}\cdot P_{3}\left (x \right )\right )\bigg | {\mstack {}{_{x \hiderel {=}0}}}=4 \\ {} & \circ & x =0\textrm {is a regular singular point}\hspace {3pt} \\ & {} & \textrm {Check to see if}\hspace {3pt} x_{0}\hspace {3pt}\textrm {is a regular singular point}\hspace {3pt} \\ {} & {} & x_{0}=0 \\ \bullet & {} & \textrm {Multiply by denominators}\hspace {3pt} \\ {} & {} & \left (\frac {d}{d x}y^{\prime }\right ) x^{2} \left (-1+x \right )-x \left (5 x -3\right ) y^{\prime }+y \left (-4+5 x \right )=0 \\ \bullet & {} & \textrm {Assume series solution for}\hspace {3pt} y \\ {} & {} & y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k +r} \\ \square & {} & \textrm {Rewrite ODE with series expansions}\hspace {3pt} \\ {} & \circ & \textrm {Convert}\hspace {3pt} x^{m}\cdot y\hspace {3pt}\textrm {to series expansion for}\hspace {3pt} m =0..1 \\ {} & {} & x^{m}\cdot y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k +r +m} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k -m \\ {} & {} & x^{m}\cdot y=\moverset {\infty }{\munderset {k =m}{\sum }}a_{k -m} x^{k +r} \\ {} & \circ & \textrm {Convert}\hspace {3pt} x^{m}\cdot y^{\prime }\hspace {3pt}\textrm {to series expansion for}\hspace {3pt} m =1..2 \\ {} & {} & x^{m}\cdot y^{\prime }=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (k +r \right ) x^{k +r -1+m} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +1-m \\ {} & {} & x^{m}\cdot y^{\prime }=\moverset {\infty }{\munderset {k =-1+m}{\sum }}a_{k +1-m} \left (k +1-m +r \right ) x^{k +r} \\ {} & \circ & \textrm {Convert}\hspace {3pt} x^{m}\cdot \left (\frac {d}{d x}y^{\prime }\right )\hspace {3pt}\textrm {to series expansion for}\hspace {3pt} m =2..3 \\ {} & {} & x^{m}\cdot \left (\frac {d}{d x}y^{\prime }\right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (k +r \right ) \left (k +r -1\right ) x^{k +r -2+m} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +2-m \\ {} & {} & x^{m}\cdot \left (\frac {d}{d x}y^{\prime }\right )=\moverset {\infty }{\munderset {k =-2+m}{\sum }}a_{k +2-m} \left (k +2-m +r \right ) \left (k +1-m +r \right ) x^{k +r} \\ & {} & \textrm {Rewrite ODE with series expansions}\hspace {3pt} \\ {} & {} & -a_{0} \left (-2+r \right )^{2} x^{r}+\left (\moverset {\infty }{\munderset {k =1}{\sum }}\left (-a_{k} \left (k +r -2\right )^{2}+a_{k -1} \left (k +r -2\right ) \left (k -6+r \right )\right ) x^{k +r}\right )=0 \\ \bullet & {} & a_{0}\textrm {cannot be 0 by assumption, giving the indicial equation}\hspace {3pt} \\ {} & {} & -\left (-2+r \right )^{2}=0 \\ \bullet & {} & \textrm {Values of r that satisfy the indicial equation}\hspace {3pt} \\ {} & {} & r =2 \\ \bullet & {} & \textrm {Each term in the series must be 0, giving the recursion relation}\hspace {3pt} \\ {} & {} & -a_{k} \left (k +r -2\right )^{2}+a_{k -1} \left (k +r -2\right ) \left (k -6+r \right )=0 \\ \bullet & {} & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +1 \\ {} & {} & -a_{k +1} \left (k +r -1\right )^{2}+a_{k} \left (k +r -1\right ) \left (k +r -5\right )=0 \\ \bullet & {} & \textrm {Recursion relation that defines series solution to ODE}\hspace {3pt} \\ {} & {} & a_{k +1}=\frac {a_{k} \left (k +r -5\right )}{k +r -1} \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =2\hspace {3pt}\textrm {; series terminates at}\hspace {3pt} k =3 \\ {} & {} & a_{k +1}=\frac {a_{k} \left (k -3\right )}{k +1} \\ \bullet & {} & \textrm {Apply recursion relation for}\hspace {3pt} k =0 \\ {} & {} & a_{1}=-3 a_{0} \\ \bullet & {} & \textrm {Apply recursion relation for}\hspace {3pt} k =1 \\ {} & {} & a_{2}=-a_{1} \\ \bullet & {} & \textrm {Express in terms of}\hspace {3pt} a_{0} \\ {} & {} & a_{2}=3 a_{0} \\ \bullet & {} & \textrm {Apply recursion relation for}\hspace {3pt} k =2 \\ {} & {} & a_{3}=-\frac {a_{2}}{3} \\ \bullet & {} & \textrm {Express in terms of}\hspace {3pt} a_{0} \\ {} & {} & a_{3}=-a_{0} \\ \bullet & {} & \textrm {Terminating series solution of the ODE for}\hspace {3pt} r =2\hspace {3pt}\textrm {. Use reduction of order to find the second linearly independent solution}\hspace {3pt} \\ {} & {} & y=a_{0}\cdot \left (-x^{3}+3 x^{2}-3 x +1\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) Group is reducible, not completely reducible <- Kovacics algorithm successful`
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 61
Order:=6; dsolve(x^2*(1-x)*diff(y(x),x$2)-x*(3-5*x)*diff(y(x),x)+(4-5*x)*y(x)=0,y(x),type='series',x=0);
\[ y \left (x \right ) = \left (\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-3 x +3 x^{2}-x^{3}+\operatorname {O}\left (x^{6}\right )\right )+\left (4 x -7 x^{2}+\frac {11}{3} x^{3}-\frac {1}{4} x^{4}-\frac {1}{20} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \right ) x^{2} \]
✓ Solution by Mathematica
Time used: 0.007 (sec). Leaf size: 84
AsymptoticDSolveValue[x^2*(1-x)*y''[x]-x*(3-5*x)*y'[x]+(4-5*x)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (-x^3+3 x^2-3 x+1\right ) x^2+c_2 \left (\left (-x^3+3 x^2-3 x+1\right ) x^2 \log (x)+\left (-\frac {x^5}{20}-\frac {x^4}{4}+\frac {11 x^3}{3}-7 x^2+4 x\right ) x^2\right ) \]