Internal problem ID [13943]
Internal file name [OUTPUT/13115_Friday_February_23_2024_06_54_54_AM_30724087/index.tex]
Book: Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell.
second edition. CRC Press. FL, USA. 2020
Section: Chapter 33. Power series solutions I: Basic computational methods. Additional Exercises.
page 641
Problem number: 33.11 (h).
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "first order ode series method. Taylor series method"
Maple gives the following as the ode type
[_quadrature]
\[ \boxed {y^{\prime }+\cos \left (y\right )=0} \] With the expansion point for the power series method at \(x = 0\).
Solving ode using Taylor series method. This gives review on how the Taylor series method works for solving first order ode. Let \[ y^{\prime }=f\left ( x,y\right ) \] Where \(f\left ( x,y\right ) \) is analytic at expansion point \(x_{0}\). We can always shift to \(x_{0}=0\) if \(x_{0}\) is not zero. So from now we assume \(x_{0}=0\,\). Assume also that \(y\left ( x_{0}\right ) =y_{0}\). Using Taylor series\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 }\left ( x_{0}\right ) +\cdots \\ & =y_{0}+xf+\frac {x^{2}}{2}\left . \frac {df}{dx}\right \vert _{x_{0},y_{0}}+\frac {x^{3}}{3!}\left . \frac {d^{2}f}{dx^{2}}\right \vert _{x_{0},y_{0}}+\cdots \\ & =y_{0}+\sum _{n=0}^{\infty }\frac {x^{n+1}}{\left ( n+1\right ) !}\left . \frac {d^{n}f}{dx^{n}}\right \vert _{x_{0},y_{0}} \end {align*}
But \begin {align} \frac {df}{dx} & =\frac {\partial f}{\partial x}+\frac {\partial f}{\partial y}f\tag {1}\\ \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 ) 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 ) f\tag {3}\\ & \vdots \nonumber \end {align}
And so on. Hence if we name \(F_{0}=f\left ( x,y\right ) \) then the above can be written as \begin {align} F_{0} & =f\left ( x,y\right ) \tag {4}\\ 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 ) F_{0} \tag {5} \end {align}
For example, for \(n=1\,\) we see that \begin {align*} F_{1} & =\frac {d}{dx}\left ( F_{0}\right ) \\ & =\frac {\partial }{\partial x}F_{0}+\left ( \frac {\partial F_{0}}{\partial y}\right ) F_{0}\\ & =\frac {\partial f}{\partial x}+\frac {\partial f}{\partial y}f \end {align*}
Which is (1). And when \(n=2\)\begin {align*} F_{2} & =\frac {d}{dx}\left ( F_{1}\right ) \\ & =\frac {\partial }{\partial x}F_{1}+\left ( \frac {\partial F_{1}}{\partial y}\right ) F_{0}\\ & =\frac {\partial }{\partial x}\left ( \frac {\partial f}{\partial x}+\frac {\partial f}{\partial y}f\right ) +\frac {\partial }{\partial y}\left ( \frac {\partial f}{\partial x}+\frac {\partial f}{\partial y}f\right ) f\\ & =\frac {\partial }{\partial x}\left ( \frac {df}{dx}\right ) +\frac {\partial }{\partial y}\left ( \frac {df}{dx}\right ) f \end {align*}
Which is (2) and so on. Therefore (4,5) can be used from now on along with \begin {equation} y\left ( x\right ) =y_{0}+\sum _{n=0}^{\infty }\frac {x^{n+1}}{\left ( n+1\right ) !}\left . F_{n}\right \vert _{x_{0},y_{0}} \tag {6} \end {equation} Hence \begin {align*} F_0 &= -\cos \left (y\right )\\ F_1 &= \frac {d F_0}{dx} \\ &= \frac {\partial F_0}{\partial x}+ \frac {\partial F_0}{\partial y} F_0 \\ &= -\frac {\sin \left (2 y\right )}{2}\\ F_2 &= \frac {d F_1}{dx} \\ &= \frac {\partial F_1}{\partial x}+ \frac {\partial F_1}{\partial y} F_1 \\ &= \cos \left (y\right ) \cos \left (2 y\right )\\ F_3 &= \frac {d F_2}{dx} \\ &= \frac {\partial F_2}{\partial x}+ \frac {\partial F_2}{\partial y} F_2 \\ &= \sin \left (y\right ) \left (6 \cos \left (y\right )^{2}-1\right ) \cos \left (y\right ) \end {align*}
And so on. Evaluating all the above at initial conditions \(x \left (0\right ) = 0\) and \(y \left (0\right ) = y \left (0\right )\) gives \begin {align*} F_0 &= -\cos \left (y \left (0\right )\right )\\ F_1 &= -\frac {\sin \left (2 y \left (0\right )\right )}{2}\\ F_2 &= \cos \left (y \left (0\right )\right ) \cos \left (2 y \left (0\right )\right )\\ F_3 &= \frac {3 \sin \left (4 y \left (0\right )\right )}{4}+\sin \left (2 y \left (0\right )\right ) \end {align*}
Substituting all the above in (6) and simplifying gives the solution as \[ y = y \left (0\right )-\cos \left (y \left (0\right )\right ) x -\frac {x^{2} \sin \left (y \left (0\right )\right ) \cos \left (y \left (0\right )\right )}{2}+\frac {\cos \left (y \left (0\right )\right )^{3} x^{3}}{3}-\frac {\cos \left (y \left (0\right )\right ) x^{3}}{6}+\frac {x^{4} \cos \left (y \left (0\right )\right )^{3} \sin \left (y \left (0\right )\right )}{4}-\frac {x^{4} \sin \left (y \left (0\right )\right ) \cos \left (y \left (0\right )\right )}{24}+O\left (x^{5}\right ) \] Unable to also solve using normal power series since not linear ode. Not currently supported.
The solution(s) found are the following \begin{align*} \tag{1} y &= y \left (0\right )-\cos \left (y \left (0\right )\right ) x -\frac {x^{2} \sin \left (y \left (0\right )\right ) \cos \left (y \left (0\right )\right )}{2}+\frac {\cos \left (y \left (0\right )\right )^{3} x^{3}}{3}-\frac {\cos \left (y \left (0\right )\right ) x^{3}}{6}+\frac {x^{4} \cos \left (y \left (0\right )\right )^{3} \sin \left (y \left (0\right )\right )}{4}-\frac {x^{4} \sin \left (y \left (0\right )\right ) \cos \left (y \left (0\right )\right )}{24}+O\left (x^{5}\right ) \\ \end{align*}
Verification of solutions
\[ y = y \left (0\right )-\cos \left (y \left (0\right )\right ) x -\frac {x^{2} \sin \left (y \left (0\right )\right ) \cos \left (y \left (0\right )\right )}{2}+\frac {\cos \left (y \left (0\right )\right )^{3} x^{3}}{3}-\frac {\cos \left (y \left (0\right )\right ) x^{3}}{6}+\frac {x^{4} \cos \left (y \left (0\right )\right )^{3} \sin \left (y \left (0\right )\right )}{4}-\frac {x^{4} \sin \left (y \left (0\right )\right ) \cos \left (y \left (0\right )\right )}{24}+O\left (x^{5}\right ) \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }=-\cos \left (y\right ) \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\cos \left (y\right ) \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{\cos \left (y\right )}=-1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{\cos \left (y\right )}d x =\int \left (-1\right )d x +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \ln \left (\sec \left (y\right )+\tan \left (y\right )\right )=-x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\arctan \left (\frac {\left ({\mathrm e}^{-x +c_{1}}\right )^{2}-1}{\left ({\mathrm e}^{-x +c_{1}}\right )^{2}+1}, \frac {2 \,{\mathrm e}^{-x +c_{1}}}{\left ({\mathrm e}^{-x +c_{1}}\right )^{2}+1}\right ) \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli trying separable <- separable successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 67
Order:=5; dsolve(diff(y(x),x)+cos(y(x))=0,y(x),type='series',x=0);
\[ y \left (x \right ) = y \left (0\right )-\cos \left (y \left (0\right )\right ) x -\frac {\sin \left (2 y \left (0\right )\right ) x^{2}}{4}+\frac {\cos \left (y \left (0\right )\right ) \cos \left (2 y \left (0\right )\right ) x^{3}}{6}+\left (\frac {\sin \left (4 y \left (0\right )\right )}{32}+\frac {\sin \left (2 y \left (0\right )\right )}{24}\right ) x^{4}+O\left (x^{5}\right ) \]
✓ Solution by Mathematica
Time used: 0.074 (sec). Leaf size: 76
AsymptoticDSolveValue[y'[x]+Cos[y[x]]==0,y[x],{x,0,4}]
\[ y(x)\to \frac {1}{24} x^4 \left (5 \sin (c_1) \cos ^3(c_1)-\sin ^3(c_1) \cos (c_1)\right )+\frac {1}{6} x^3 \left (\cos ^3(c_1)-\sin ^2(c_1) \cos (c_1)\right )-\frac {1}{2} x^2 \sin (c_1) \cos (c_1)-x \cos (c_1)+c_1 \]