Taylor Polynomial Approximations of IVPs

The foruma for the Taylor polynomial of degree $n$ centered at $x_0$, approximating a function $f(x)$ possessing $n$ derivatives at $x_0$ is given by:

$$\begin{array}{cc}{p}_n(x)=f(x_0)+f^{\prime }(x_0)(x-x_0)+\frac{f^{″}(x_0)}{2!}(x-x_0{)}^2+\frac{f^{‴}(x_0)}{3!}(x-x_0{)}^3+\cdots +\frac{f^{(n)}(x_0)}{n!}(x-x_0{)}^n=\sum _{j=0}^n\frac{f^{(j)}(x_0)}{j!}(x-x_0{)}^j.& \text{(1)}\end{array}$$

Note that the terms in the series involve derivatives of $f$, and if $x_0=0$, they involve intial values of derivatives of $f$.

Now, given a differential equation and some initial values, we can exploit this to find a Taylor polynomial approximating the solution of the IVP.

Example:

Find the first few Taylor polynomials approximating the solution around $x_0=0$ of the initial value problem

$$y^{″}=3y^{\prime }+x^{2}y;y(0)=10,y^{\prime }(0^{\prime })=5$$

In order to construct

$$\begin{array}{cc}{p}_n(x)=y(0)+y^{\prime }(0)x+\frac{y^{″}(0)}{2!}{x}^2+\frac{y^{‴}(0)}{3!}{x}^3+\cdots +\frac{y^{(n)}(0)}{n!}{x}^n,& \text{(2)}\end{array}$$

we need the values of $y(0)$, $y^{\prime }(0)$, $y^{″}(0)$, $y^{‴}(0),$ etc. The first two are given in the initial conditions, and we can deduce more by using the diffrential equation and the provided initial conditions:

$$y^{″}(0)=3y^{\prime }(0)+0^{2}y(0)=3\cdot 5+0\cdot 10=15.$$

We know that y''= 3y' = x^2y for some interval around $x_0=0$ (why?), and can differentiate both sides to derive:

$$y^{‴}=3y^{″}+2xy+x^2{y}^{\prime },$$

$$y^{(4)}=3y^{‴}+2y+4x{y}^{\prime }+x^2{y}^{″},$$

and so on as long as we wish to and as long as additional derivatives of $y$ exist.

Now, we can substitute in initial values of lower derivatives and $x=0$ to find additional initial values:

$$y^{‴}(0)=3\cdot 15+2\cdot 0\cdot 10+0^2\cdot 5=45,$$

$$y^{(4)}(0)=3\cdot 45+2\cdot 10+4\cdot 0\cdot 5+0^2+15=155,$$

and so on.

Substituting these initial conditions into (2) and dropping higher order terms we get:

$$p_4(x)=10+5x+\frac{15}{2!}{x}^2+\frac{45}{3!}{x}^3+\frac{155}{4!}{x}^4,$$

which is the Taylor polynomial of degree 4 approximating the solution to our IVP at $x_0=0$.

Note that this can be applied around places other than $x_0=0$ by translating coordinates, but I don't cover that here.

In general this process is very tedious and can be difficult to do by hand without making errors. I have written a sage script that finds the Taylor polynomial approximation for IVPs, which makes this process quick and accurate.