2.4.16 problem 16

Internal problem ID [8331]
Book : Own collection of miscellaneous problems
Section : section 4.0
Problem number : 16
Date solved : Sunday, November 10, 2024 at 03:38:38 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

Solve

\begin{align*} \left (x +1\right ) \left (3 x -1\right ) y^{\prime \prime }+\cos \left (x \right ) y^{\prime }-3 x y&=0 \end{align*}

Using series expansion around \(x=0\)

Solving ode using Taylor series method. This gives review on how the Taylor series method works for solving second order ode.

Let

\[ y^{\prime \prime }=f\left ( x,y,y^{\prime }\right ) \]

Assuming expansion is at \(x_{0}=0\) (we can always shift the actual expansion point to \(0\) by change of variables) and assuming \(f\left ( x,y,y^{\prime }\right ) \) is analytic at \(x_{0}\) which must be the case for an ordinary point. Let initial conditions be \(y\left ( x_{0}\right ) =y_{0}\) and \(y^{\prime }\left ( x_{0}\right ) =y_{0}^{\prime }\). Using Taylor series gives

\begin{align*} y\left ( x\right ) & =y\left ( x_{0}\right ) +\left ( x-x_{0}\right ) y^{\prime }\left ( x_{0}\right ) +\frac {\left ( x-x_{0}\right ) ^{2}}{2}y^{\prime \prime }\left ( x_{0}\right ) +\frac {\left ( x-x_{0}\right ) ^{3}}{3!}y^{\prime \prime \prime }\left ( x_{0}\right ) +\cdots \\ & =y_{0}+xy_{0}^{\prime }+\frac {x^{2}}{2}\left . f\right \vert _{x_{0},y_{0},y_{0}^{\prime }}+\frac {x^{3}}{3!}\left . f^{\prime }\right \vert _{x_{0},y_{0},y_{0}^{\prime }}+\cdots \\ & =y_{0}+xy_{0}^{\prime }+\sum _{n=0}^{\infty }\frac {x^{n+2}}{\left ( n+2\right ) !}\left . \frac {d^{n}f}{dx^{n}}\right \vert _{x_{0},y_{0},y_{0}^{\prime }}\end{align*}

But

\begin{align} \frac {df}{dx} & =\frac {\partial f}{\partial x}\frac {dx}{dx}+\frac {\partial f}{\partial y}\frac {dy}{dx}+\frac {\partial f}{\partial y^{\prime }}\frac {dy^{\prime }}{dx}\tag {1}\\ & =\frac {\partial f}{\partial x}+\frac {\partial f}{\partial y}y^{\prime }+\frac {\partial f}{\partial y^{\prime }}y^{\prime \prime }\\ & =\frac {\partial f}{\partial x}+\frac {\partial f}{\partial y}y^{\prime }+\frac {\partial f}{\partial y^{\prime }}f\\ \frac {d^{2}f}{dx^{2}} & =\frac {d}{dx}\left ( \frac {df}{dx}\right ) \nonumber \\ & =\frac {\partial }{\partial x}\left ( \frac {df}{dx}\right ) +\frac {\partial }{\partial y}\left ( \frac {df}{dx}\right ) y^{\prime }+\frac {\partial }{\partial y^{\prime }}\left ( \frac {df}{dx}\right ) f\tag {2}\\ \frac {d^{3}f}{dx^{3}} & =\frac {d}{dx}\left ( \frac {d^{2}f}{dx^{2}}\right ) \nonumber \\ & =\frac {\partial }{\partial x}\left ( \frac {d^{2}f}{dx^{2}}\right ) +\left ( \frac {\partial }{\partial y}\frac {d^{2}f}{dx^{2}}\right ) y^{\prime }+\frac {\partial }{\partial y^{\prime }}\left ( \frac {d^{2}f}{dx^{2}}\right ) f\tag {3}\\ & \vdots \nonumber \end{align}

And so on. Hence if we name \(F_{0}=f\left ( x,y,y^{\prime }\right ) \) then the above can be written as

\begin{align} F_{0} & =f\left ( x,y,y^{\prime }\right ) \tag {4}\\ F_{1} & =\frac {df}{dx}\nonumber \\ & =\frac {dF_{0}}{dx}\nonumber \\ & =\frac {\partial f}{\partial x}+\frac {\partial f}{\partial y}y^{\prime }+\frac {\partial f}{\partial y^{\prime }}y^{\prime \prime }\nonumber \\ & =\frac {\partial f}{\partial x}+\frac {\partial f}{\partial y}y^{\prime }+\frac {\partial f}{\partial y^{\prime }}f\tag {5}\\ & =\frac {\partial F_{0}}{\partial x}+\frac {\partial F_{0}}{\partial y}y^{\prime }+\frac {\partial F_{0}}{\partial y^{\prime }}F_{0}\nonumber \\ F_{2} & =\frac {d}{dx}\left ( \frac {d}{dx}f\right ) \nonumber \\ & =\frac {d}{dx}\left ( F_{1}\right ) \nonumber \\ & =\frac {\partial }{\partial x}F_{1}+\left ( \frac {\partial F_{1}}{\partial y}\right ) y^{\prime }+\left ( \frac {\partial F_{1}}{\partial y^{\prime }}\right ) y^{\prime \prime }\nonumber \\ & =\frac {\partial }{\partial x}F_{1}+\left ( \frac {\partial F_{1}}{\partial y}\right ) y^{\prime }+\left ( \frac {\partial F_{1}}{\partial y^{\prime }}\right ) F_{0}\nonumber \\ & \vdots \nonumber \\ F_{n} & =\frac {d}{dx}\left ( F_{n-1}\right ) \nonumber \\ & =\frac {\partial }{\partial x}F_{n-1}+\left ( \frac {\partial F_{n-1}}{\partial y}\right ) y^{\prime }+\left ( \frac {\partial F_{n-1}}{\partial y^{\prime }}\right ) y^{\prime \prime }\nonumber \\ & =\frac {\partial }{\partial x}F_{n-1}+\left ( \frac {\partial F_{n-1}}{\partial y}\right ) y^{\prime }+\left ( \frac {\partial F_{n-1}}{\partial y^{\prime }}\right ) F_{0} \tag {6}\end{align}

Therefore (6) can be used from now on along with

\begin{equation} y\left ( x\right ) =y_{0}+xy_{0}^{\prime }+\sum _{n=0}^{\infty }\frac {x^{n+2}}{\left ( n+2\right ) !}\left . F_{n}\right \vert _{x_{0},y_{0},y_{0}^{\prime }} \tag {7}\end{equation}

To find \(y\left ( x\right ) \) series solution around \(x=0\). Hence

\begin{align*} F_0 &= -\frac {\cos \left (x \right ) y^{\prime }-3 x y}{\left (x +1\right ) \left (3 x -1\right )}\\ F_1 &= \frac {d F_0}{dx} \\ &= \frac {\partial F_{0}}{\partial x}+ \frac {\partial F_{0}}{\partial y} y^{\prime }+ \frac {\partial F_{0}}{\partial y^{\prime }} F_0 \\ &= \frac {\left (\cos \left (x \right )^{2}+\left (6 x +2\right ) \cos \left (x \right )+9 \left (x +\frac {\sin \left (x \right )}{3}\right ) \left (x +1\right ) \left (x -\frac {1}{3}\right )\right ) y^{\prime }-9 y x^{2}-3 \cos \left (x \right ) y x -3 y}{\left (x +1\right )^{2} \left (3 x -1\right )^{2}}\\ F_2 &= \frac {d F_1}{dx} \\ &= \frac {\partial F_{1}}{\partial x}+ \frac {\partial F_{1}}{\partial y} y^{\prime }+ \frac {\partial F_{1}}{\partial y^{\prime }} F_1 \\ &= \frac {\left (-\cos \left (x \right )^{3}+\left (-18 x -6\right ) \cos \left (x \right )^{2}+\left (\left (-9 x^{2}-6 x +3\right ) \sin \left (x \right )+9 x^{4}-6 x^{3}-68 x^{2}-34 x -13\right ) \cos \left (x \right )-54 \left (\left (\frac {2 x}{3}+\frac {2}{9}\right ) \sin \left (x \right )+x^{2}+\frac {1}{3}\right ) \left (x +1\right ) \left (x -\frac {1}{3}\right )\right ) y^{\prime }+27 y \left (\frac {\cos \left (x \right )^{2} x}{9}+\left (\frac {5}{3} x^{2}+\frac {4}{9} x +\frac {1}{9}\right ) \cos \left (x \right )+\left (\frac {2}{3} x^{3}+\frac {4}{9} x^{2}-\frac {2}{9} x \right ) \sin \left (x \right )+x^{4}+\frac {8 x^{3}}{3}-\frac {x^{2}}{3}+2 x +\frac {4}{9}\right )}{\left (x +1\right )^{3} \left (3 x -1\right )^{3}}\\ F_3 &= \frac {d F_2}{dx} \\ &= \frac {\partial F_{2}}{\partial x}+ \frac {\partial F_{2}}{\partial y} y^{\prime }+ \frac {\partial F_{2}}{\partial y^{\prime }} F_2 \\ &= \frac {\left (\cos \left (x \right )^{4}+\left (36 x +12\right ) \cos \left (x \right )^{3}+\left (\left (18 x^{2}+12 x -6\right ) \sin \left (x \right )-63 x^{4}-57 x^{3}+356 x^{2}+235 x +61\right ) \cos \left (x \right )^{2}+\left (\left (252 x^{3}+252 x^{2}-28 x -28\right ) \sin \left (x \right )-162 x^{5}+81 x^{4}+918 x^{3}+708 x^{2}+516 x +99\right ) \cos \left (x \right )-27 \left (\left (x^{4}-\frac {8}{3} x^{3}-\frac {188}{9} x^{2}-\frac {100}{9} x -\frac {41}{9}\right ) \sin \left (x \right )-3 x^{4}-20 x^{3}-\frac {56 x}{3}-\frac {11}{3}\right ) \left (x +1\right ) \left (x -\frac {1}{3}\right )\right ) y^{\prime }+81 y \left (-\frac {x \cos \left (x \right )^{3}}{27}+\left (-\frac {11}{9} x^{2}-\frac {10}{27} x -\frac {1}{27}\right ) \cos \left (x \right )^{2}+\left (\left (-\frac {5}{9} x^{3}-\frac {10}{27} x^{2}+\frac {5}{27} x \right ) \sin \left (x \right )+x^{5}+\frac {2 x^{4}}{3}-\frac {28 x^{3}}{3}-\frac {44 x^{2}}{9}-\frac {25 x}{9}-\frac {10}{27}\right ) \cos \left (x \right )+\left (-5 x^{4}-\frac {14}{3} x^{3}+\frac {4}{9} x^{2}+\frac {2}{9} x +\frac {1}{9}\right ) \sin \left (x \right )-4 x^{5}-\frac {26 x^{4}}{3}-\frac {116 x^{2}}{9}-\frac {44 x}{9}-\frac {14}{9}\right )}{\left (x +1\right )^{4} \left (3 x -1\right )^{4}}\\ F_4 &= \frac {d F_3}{dx} \\ &= \frac {\partial F_{3}}{\partial x}+ \frac {\partial F_{3}}{\partial y} y^{\prime }+ \frac {\partial F_{3}}{\partial y^{\prime }} F_3 \\ &= \frac {\left (-\cos \left (x \right )^{5}+\left (-60 x -20\right ) \cos \left (x \right )^{4}+\left (\left (-30 x^{2}-20 x +10\right ) \sin \left (x \right )+225 x^{4}+264 x^{3}-1154 x^{2}-808 x -175\right ) \cos \left (x \right )^{3}+\left (\left (-900 x^{3}-900 x^{2}+100 x +100\right ) \sin \left (x \right )+2430 x^{5}+2997 x^{4}-6912 x^{3}-7650 x^{2}-4134 x -763\right ) \cos \left (x \right )^{2}+\left (\left (405 x^{6}+405 x^{5}-6615 x^{4}-8610 x^{3}-1845 x^{2}+565 x +415\right ) \sin \left (x \right )-81 x^{8}+351 x^{7}+3699 x^{6}-987 x^{5}-15067 x^{4}-16519 x^{3}-17271 x^{2}-6637 x -1216\right ) \cos \left (x \right )+648 \left (x -\frac {1}{3}\right ) \left (\left (x^{5}-\frac {10}{3} x^{4}-\frac {293}{18} x^{3}-\frac {655}{54} x^{2}-\frac {503}{54} x -\frac {109}{54}\right ) \sin \left (x \right )-\frac {9 x^{5}}{4}-\frac {21 x^{4}}{2}-\frac {5 x^{3}}{3}-\frac {121 x^{2}}{6}-\frac {817 x}{108}-\frac {58}{27}\right ) \left (x +1\right )\right ) y^{\prime }-324 \left (-\frac {\cos \left (x \right )^{4} x}{108}+\left (-\frac {19}{36} x^{2}-\frac {1}{6} x -\frac {1}{108}\right ) \cos \left (x \right )^{3}+\left (\left (-\frac {1}{4} x^{3}-\frac {1}{6} x^{2}+\frac {1}{12} x \right ) \sin \left (x \right )+\frac {59 x^{4}}{36}-\frac {485 x^{3}}{54}-\frac {617 x^{2}}{108}-\frac {67 x}{36}+\frac {17 x^{5}}{12}-\frac {1}{6}\right ) \cos \left (x \right )^{2}+\left (\left (\frac {7}{108}-\frac {25}{4} x^{4}-\frac {109}{18} x^{3}+\frac {17}{27} x^{2}+\frac {1}{2} x \right ) \sin \left (x \right )+\frac {27 x^{5}}{4}-\frac {83 x^{4}}{2}-\frac {307 x^{3}}{9}-\frac {94 x^{2}}{3}-\frac {39 x}{4}+\frac {15 x^{6}}{2}-\frac {38}{27}\right ) \cos \left (x \right )+\left (x^{5}-\frac {1}{6} x^{4}-\frac {245}{9} x^{3}-\frac {251}{18} x^{2}-\frac {77}{9} x -\frac {10}{9}\right ) \left (x -\frac {1}{3}\right ) \left (x +1\right ) \sin \left (x \right )-\frac {13}{3}-\frac {29 x^{6}}{2}-\frac {3 x^{7}}{4}-\frac {55 x^{5}}{2}-\frac {77 x^{4}}{9}-\frac {7625 x^{3}}{108}-\frac {2021 x^{2}}{54}-\frac {617 x}{27}\right ) y}{\left (x +1\right )^{5} \left (3 x -1\right )^{5}} \end{align*}

And so on. Evaluating all the above at initial conditions \(x = 0\) and \(y \left (0\right ) = y \left (0\right )\) and \(y^{\prime }\left (0\right ) = y^{\prime }\left (0\right )\) gives

\begin{align*} F_0 &= y^{\prime }\left (0\right )\\ F_1 &= 3 y^{\prime }\left (0\right )-3 y \left (0\right )\\ F_2 &= 14 y^{\prime }\left (0\right )-15 y \left (0\right )\\ F_3 &= 140 y^{\prime }\left (0\right )-159 y \left (0\right )\\ F_4 &= 1711 y^{\prime }\left (0\right )-1917 y \left (0\right ) \end{align*}

Substituting all the above in (7) and simplifying gives the solution as

\[ y = \left (1-\frac {1}{2} x^{3}-\frac {5}{8} x^{4}-\frac {53}{40} x^{5}\right ) y \left (0\right )+\left (x +\frac {1}{2} x^{2}+\frac {1}{2} x^{3}+\frac {7}{12} x^{4}+\frac {7}{6} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

Since the expansion point \(x = 0\) is an ordinary point, then this can also be solved using the standard power series method. The ode is normalized to be

\[ \left (3 x^{2}+2 x -1\right ) y^{\prime \prime }+\cos \left (x \right ) y^{\prime }-3 x y = 0 \]

Let the solution be represented as power series of the form

\[ y = \moverset {\infty }{\munderset {n =0}{\sum }}a_{n} x^{n} \]

Then

\begin{align*} y^{\prime } &= \moverset {\infty }{\munderset {n =1}{\sum }}n a_{n} x^{n -1}\\ y^{\prime \prime } &= \moverset {\infty }{\munderset {n =2}{\sum }}n \left (n -1\right ) a_{n} x^{n -2} \end{align*}

Substituting the above back into the ode gives

\begin{align*} \left (3 x^{2}+2 x -1\right ) \left (\moverset {\infty }{\munderset {n =2}{\sum }}n \left (n -1\right ) a_{n} x^{n -2}\right )+\cos \left (x \right ) \left (\moverset {\infty }{\munderset {n =1}{\sum }}n a_{n} x^{n -1}\right )-3 x \left (\moverset {\infty }{\munderset {n =0}{\sum }}a_{n} x^{n}\right ) = 0\tag {1} \end{align*}

Expanding \(\cos \left (x \right )\) as Taylor series around \(x=0\) and keeping only the first \(6\) terms gives

\begin{align*} \cos \left (x \right ) &= 1-\frac {1}{2} x^{2}+\frac {1}{24} x^{4}-\frac {1}{720} x^{6} + \dots \\ &= 1-\frac {1}{2} x^{2}+\frac {1}{24} x^{4}-\frac {1}{720} x^{6} \end{align*}

Hence the ODE in Eq (1) becomes

\[ \left (3 x^{2}+2 x -1\right ) \left (\moverset {\infty }{\munderset {n =2}{\sum }}n \left (n -1\right ) a_{n} x^{n -2}\right )+\left (1-\frac {1}{2} x^{2}+\frac {1}{24} x^{4}-\frac {1}{720} x^{6}\right ) \left (\moverset {\infty }{\munderset {n =1}{\sum }}n a_{n} x^{n -1}\right )-3 x \left (\moverset {\infty }{\munderset {n =0}{\sum }}a_{n} x^{n}\right ) = 0 \]

Expanding the second term in (1) gives

\[ \left (3 x^{2}+2 x -1\right ) \left (\moverset {\infty }{\munderset {n =2}{\sum }}n \left (n -1\right ) a_{n} x^{n -2}\right )+\left (1\eslowast \left (\moverset {\infty }{\munderset {n =1}{\sum }}n a_{n} x^{n -1}\right )\right )-\frac {x^{2}}{2}\eslowast \left (\moverset {\infty }{\munderset {n =1}{\sum }}n a_{n} x^{n -1}\right )+\frac {x^{4}}{24}\eslowast \left (\moverset {\infty }{\munderset {n =1}{\sum }}n a_{n} x^{n -1}\right )-\frac {x^{6}}{720}\eslowast \left (\moverset {\infty }{\munderset {n =1}{\sum }}n a_{n} x^{n -1}\right )-3 x \left (\moverset {\infty }{\munderset {n =0}{\sum }}a_{n} x^{n}\right ) = 0 \]

Which simplifies to

\begin{equation} \tag{2} \left (\moverset {\infty }{\munderset {n =2}{\sum }}3 x^{n} a_{n} n \left (n -1\right )\right )+\left (\moverset {\infty }{\munderset {n =2}{\sum }}2 n \,x^{n -1} a_{n} \left (n -1\right )\right )+\moverset {\infty }{\munderset {n =2}{\sum }}\left (-n \left (n -1\right ) a_{n} x^{n -2}\right )+\left (\moverset {\infty }{\munderset {n =1}{\sum }}n a_{n} x^{n -1}\right )+\moverset {\infty }{\munderset {n =1}{\sum }}\left (-\frac {n \,x^{1+n} a_{n}}{2}\right )+\left (\moverset {\infty }{\munderset {n =1}{\sum }}\frac {n \,x^{n +3} a_{n}}{24}\right )+\moverset {\infty }{\munderset {n =1}{\sum }}\left (-\frac {n \,x^{n +5} a_{n}}{720}\right )+\moverset {\infty }{\munderset {n =0}{\sum }}\left (-3 x^{1+n} a_{n}\right ) = 0 \end{equation}

The next step is to make all powers of \(x\) be \(n\) in each summation term. Going over each summation term above with power of \(x\) in it which is not already \(x^{n}\) and adjusting the power and the corresponding index gives

\begin{align*} \moverset {\infty }{\munderset {n =2}{\sum }}2 n \,x^{n -1} a_{n} \left (n -1\right ) &= \moverset {\infty }{\munderset {n =1}{\sum }}2 \left (1+n \right ) a_{1+n} n \,x^{n} \\ \moverset {\infty }{\munderset {n =2}{\sum }}\left (-n \left (n -1\right ) a_{n} x^{n -2}\right ) &= \moverset {\infty }{\munderset {n =0}{\sum }}\left (-\left (n +2\right ) a_{n +2} \left (1+n \right ) x^{n}\right ) \\ \moverset {\infty }{\munderset {n =1}{\sum }}n a_{n} x^{n -1} &= \moverset {\infty }{\munderset {n =0}{\sum }}\left (1+n \right ) a_{1+n} x^{n} \\ \moverset {\infty }{\munderset {n =1}{\sum }}\left (-\frac {n \,x^{1+n} a_{n}}{2}\right ) &= \moverset {\infty }{\munderset {n =2}{\sum }}\left (-\frac {\left (n -1\right ) a_{n -1} x^{n}}{2}\right ) \\ \moverset {\infty }{\munderset {n =1}{\sum }}\frac {n \,x^{n +3} a_{n}}{24} &= \moverset {\infty }{\munderset {n =4}{\sum }}\frac {\left (n -3\right ) a_{n -3} x^{n}}{24} \\ \moverset {\infty }{\munderset {n =1}{\sum }}\left (-\frac {n \,x^{n +5} a_{n}}{720}\right ) &= \moverset {\infty }{\munderset {n =6}{\sum }}\left (-\frac {\left (n -5\right ) a_{n -5} x^{n}}{720}\right ) \\ \moverset {\infty }{\munderset {n =0}{\sum }}\left (-3 x^{1+n} a_{n}\right ) &= \moverset {\infty }{\munderset {n =1}{\sum }}\left (-3 a_{n -1} x^{n}\right ) \\ \end{align*}

Substituting all the above in Eq (2) gives the following equation where now all powers of \(x\) are the same and equal to \(n\).

\begin{equation} \tag{3} \left (\moverset {\infty }{\munderset {n =2}{\sum }}3 x^{n} a_{n} n \left (n -1\right )\right )+\left (\moverset {\infty }{\munderset {n =1}{\sum }}2 \left (1+n \right ) a_{1+n} n \,x^{n}\right )+\moverset {\infty }{\munderset {n =0}{\sum }}\left (-\left (n +2\right ) a_{n +2} \left (1+n \right ) x^{n}\right )+\left (\moverset {\infty }{\munderset {n =0}{\sum }}\left (1+n \right ) a_{1+n} x^{n}\right )+\moverset {\infty }{\munderset {n =2}{\sum }}\left (-\frac {\left (n -1\right ) a_{n -1} x^{n}}{2}\right )+\left (\moverset {\infty }{\munderset {n =4}{\sum }}\frac {\left (n -3\right ) a_{n -3} x^{n}}{24}\right )+\moverset {\infty }{\munderset {n =6}{\sum }}\left (-\frac {\left (n -5\right ) a_{n -5} x^{n}}{720}\right )+\moverset {\infty }{\munderset {n =1}{\sum }}\left (-3 a_{n -1} x^{n}\right ) = 0 \end{equation}

\(n=0\) gives

\[ -2 a_{2}+a_{1}=0 \]

\[ a_{2} = \frac {a_{1}}{2} \]

\(n=1\) gives

\[ 6 a_{2}-6 a_{3}-3 a_{0}=0 \]

Which after substituting earlier equations, simplifies to

\[ a_{3} = -\frac {a_{0}}{2}+\frac {a_{1}}{2} \]

\(n=2\) gives

\[ 6 a_{2}+15 a_{3}-12 a_{4}-\frac {7 a_{1}}{2}=0 \]

Which after substituting earlier equations, simplifies to

\[ a_{4} = -\frac {5 a_{0}}{8}+\frac {7 a_{1}}{12} \]

\(n=3\) gives

\[ 18 a_{3}+28 a_{4}-20 a_{5}-4 a_{2} = 0 \]

Which after substituting earlier equations, simplifies to

\[ -\frac {53 a_{0}}{2}+\frac {70 a_{1}}{3}-20 a_{5} = 0 \]

Or

\[ a_{5} = -\frac {53 a_{0}}{40}+\frac {7 a_{1}}{6} \]

\(n=4\) gives

\[ 36 a_{4}+45 a_{5}-30 a_{6}-\frac {9 a_{3}}{2}+\frac {a_{1}}{24}=0 \]

Which after substituting earlier equations, simplifies to

\[ a_{6} = -\frac {213 a_{0}}{80}+\frac {1711 a_{1}}{720} \]

\(n=5\) gives

\[ 60 a_{5}+66 a_{6}-42 a_{7}-5 a_{4}+\frac {a_{2}}{12} = 0 \]

Which after substituting earlier equations, simplifies to

\[ -\frac {2521 a_{0}}{10}+\frac {6719 a_{1}}{30}-42 a_{7} = 0 \]

Or

\[ a_{7} = -\frac {2521 a_{0}}{420}+\frac {6719 a_{1}}{1260} \]

For \(6\le n\), the recurrence equation is

\begin{equation} \tag{4} 3 n a_{n} \left (n -1\right )+2 \left (1+n \right ) a_{1+n} n -\left (n +2\right ) a_{n +2} \left (1+n \right )+\left (1+n \right ) a_{1+n}-\frac {\left (n -1\right ) a_{n -1}}{2}+\frac {\left (n -3\right ) a_{n -3}}{24}-\frac {\left (n -5\right ) a_{n -5}}{720}-3 a_{n -1} = 0 \end{equation}

Solving for \(a_{n +2}\), gives

\begin{align*} \tag{5} a_{n +2}&= \frac {2160 n^{2} a_{n}+1440 n^{2} a_{1+n}-2160 n a_{n}+2160 n a_{1+n}-n a_{n -5}+30 n a_{n -3}-360 n a_{n -1}+720 a_{1+n}+5 a_{n -5}-90 a_{n -3}-1800 a_{n -1}}{720 \left (n +2\right ) \left (1+n \right )} \\ &= \frac {\left (2160 n^{2}-2160 n \right ) a_{n}}{720 \left (n +2\right ) \left (1+n \right )}+\frac {\left (1440 n^{2}+2160 n +720\right ) a_{1+n}}{720 \left (n +2\right ) \left (1+n \right )}+\frac {\left (-n +5\right ) a_{n -5}}{720 \left (n +2\right ) \left (1+n \right )}+\frac {\left (30 n -90\right ) a_{n -3}}{720 \left (n +2\right ) \left (1+n \right )}+\frac {\left (-360 n -1800\right ) a_{n -1}}{720 \left (n +2\right ) \left (1+n \right )} \\ \end{align*}

And so on. Therefore the solution is

\begin{align*} y &= \moverset {\infty }{\munderset {n =0}{\sum }}a_{n} x^{n}\\ &= a_{3} x^{3}+a_{2} x^{2}+a_{1} x +a_{0} + \dots \end{align*}

Substituting the values for \(a_{n}\) found above, the solution becomes

\[ y = a_{0}+a_{1} x +\frac {a_{1} x^{2}}{2}+\left (-\frac {a_{0}}{2}+\frac {a_{1}}{2}\right ) x^{3}+\left (-\frac {5 a_{0}}{8}+\frac {7 a_{1}}{12}\right ) x^{4}+\left (-\frac {53 a_{0}}{40}+\frac {7 a_{1}}{6}\right ) x^{5}+\dots \]

Collecting terms, the solution becomes

\begin{equation} \tag{3} y = \left (1-\frac {1}{2} x^{3}-\frac {5}{8} x^{4}-\frac {53}{40} x^{5}\right ) a_{0}+\left (x +\frac {1}{2} x^{2}+\frac {1}{2} x^{3}+\frac {7}{12} x^{4}+\frac {7}{6} x^{5}\right ) a_{1}+O\left (x^{6}\right ) \end{equation}

At \(x = 0\) the solution above becomes

\[ y = \left (1-\frac {1}{2} x^{3}-\frac {5}{8} x^{4}-\frac {53}{40} x^{5}\right ) c_1 +\left (x +\frac {1}{2} x^{2}+\frac {1}{2} x^{3}+\frac {7}{12} x^{4}+\frac {7}{6} x^{5}\right ) c_2 +O\left (x^{6}\right ) \]

Maple step by step solution

Maple trace

`Methods for second order ODEs: 
--- Trying classification methods --- 
trying a symmetry of the form [xi=0, eta=F(x)] 
checking if the LODE is missing y 
-> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius 
-> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(int(r(x), dx)) * 2F1([a1, a2], [b1], f) 
-> Trying changes of variables to rationalize or make the ODE simpler 
   trying a symmetry of the form [xi=0, eta=F(x)] 
   checking if the LODE is missing y 
   -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius 
   -> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(int(r(x), dx)) * 2F1([a1, a2], [b1], f) 
      trying a symmetry of the form [xi=0, eta=F(x)] 
      trying 2nd order exact linear 
      trying symmetries linear in x and y(x) 
      trying to convert to a linear ODE with constant coefficients 
      -> trying with_periodic_functions in the coefficients 
         --- Trying Lie symmetry methods, 2nd order --- 
         `, `-> Computing symmetries using: way = 5 
   trying a symmetry of the form [xi=0, eta=F(x)] 
   checking if the LODE is missing y 
   -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius 
   -> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(int(r(x), dx)) * 2F1([a1, a2], [b1], f) 
      trying a symmetry of the form [xi=0, eta=F(x)] 
      trying 2nd order exact linear 
      trying symmetries linear in x and y(x) 
      trying to convert to a linear ODE with constant coefficients 
<- unable to find a useful change of variables 
   trying a symmetry of the form [xi=0, eta=F(x)] 
   trying differential order: 2; exact nonlinear 
   trying symmetries linear in x and y(x) 
   trying to convert to a linear ODE with constant coefficients 
   trying 2nd order, integrating factor of the form mu(x,y) 
   trying a symmetry of the form [xi=0, eta=F(x)] 
   checking if the LODE is missing y 
   -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius 
   -> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(int(r(x), dx)) * 2F1([a1, a2], [b1], f) 
   -> Trying changes of variables to rationalize or make the ODE simpler 
      trying a symmetry of the form [xi=0, eta=F(x)] 
      checking if the LODE is missing y 
      -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius 
      -> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(int(r(x), dx)) * 2F1([a1, a2], [b1], f) 
         trying a symmetry of the form [xi=0, eta=F(x)] 
         trying 2nd order exact linear 
         trying symmetries linear in x and y(x) 
         trying to convert to a linear ODE with constant coefficients 
         -> trying with_periodic_functions in the coefficients 
      trying a symmetry of the form [xi=0, eta=F(x)] 
      checking if the LODE is missing y 
      -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius 
      -> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(int(r(x), dx)) * 2F1([a1, a2], [b1], f) 
         trying a symmetry of the form [xi=0, eta=F(x)] 
         trying 2nd order exact linear 
         trying symmetries linear in x and y(x) 
         trying to convert to a linear ODE with constant coefficients 
         -> trying with_periodic_functions in the coefficients 
      trying a symmetry of the form [xi=0, eta=F(x)] 
      checking if the LODE is missing y 
      -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius 
      -> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(int(r(x), dx)) * 2F1([a1, a2], [b1], f) 
         trying a symmetry of the form [xi=0, eta=F(x)] 
         trying 2nd order exact linear 
         trying symmetries linear in x and y(x) 
         trying to convert to a linear ODE with constant coefficients 
   <- unable to find a useful change of variables 
      trying a symmetry of the form [xi=0, eta=F(x)] 
   trying to convert to an ODE of Bessel type 
   -> trying reduction of order to Riccati 
      trying Riccati sub-methods: 
         trying Riccati_symmetries 
         -> trying a symmetry pattern of the form [F(x)*G(y), 0] 
         -> trying a symmetry pattern of the form [0, F(x)*G(y)] 
         -> trying a symmetry pattern of the form [F(x),G(x)*y+H(x)] 
   -> trying with_periodic_functions in the coefficients 
      --- Trying Lie symmetry methods, 2nd order --- 
      `, `-> Computing symmetries using: way = 5 
--- Trying Lie symmetry methods, 2nd order --- 
`, `-> Computing symmetries using: way = 3`[0, y]
 

Maple dsolve solution

Solving time : 0.007 (sec)
Leaf size : 54

dsolve((x+1)*(3*x-1)*diff(diff(y(x),x),x)+diff(y(x),x)*cos(x)-3*x*y(x) = 0,y(x), 
       series,x=0)
 

\[ y = \left (1-\frac {1}{2} x^{3}-\frac {5}{8} x^{4}-\frac {53}{40} x^{5}\right ) y \left (0\right )+\left (x +\frac {1}{2} x^{2}+\frac {1}{2} x^{3}+\frac {7}{12} x^{4}+\frac {7}{6} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

Mathematica DSolve solution

Solving time : 0.004 (sec)
Leaf size : 63

AsymptoticDSolveValue[{(x+1)*(3*x-1)*D[y[x],{x,2}]+Cos[x]*D[y[x],x]-3*x*y[x]==0,{}}, 
       y[x],{x,0,5}]
 

\[ y(x)\to c_1 \left (-\frac {53 x^5}{40}-\frac {5 x^4}{8}-\frac {x^3}{2}+1\right )+c_2 \left (\frac {7 x^5}{6}+\frac {7 x^4}{12}+\frac {x^3}{2}+\frac {x^2}{2}+x\right ) \]