Consider the differential equation $\frac{dy}{dx} = 6x^{2} - x^{2}y$. Let $y = f(x)$ be a particular solution to this differential equation with the initial condition $f(-1) = 2$. (a) Use Euler's method with two steps of equal size, starting at $x = -1$, to approximate $f(0)$. Show the work that leads to your answer. (b) At the point $(-1, 2)$, the value of $\frac{d^{2}y}{dx^{2}}$ is $-12$. Find the second-degree Taylor polynomial for $f$ about $x = -1$. (c) Find the particular solution $y = f(x)$ to the given differential equation with the initial condition $f(-1) = 2$.
Consider the differential equation $\frac{dy}{dx} = 6x^{2} - x^{2}y$. Let $y = f(x)$ be a particular solution to this differential equation with the initial condition $f(-1) = 2$.
(a) Use Euler's method with two steps of equal size, starting at $x = -1$, to approximate $f(0)$. Show the work that leads to your answer.
(b) At the point $(-1, 2)$, the value of $\frac{d^{2}y}{dx^{2}}$ is $-12$. Find the second-degree Taylor polynomial for $f$ about $x = -1$.
(c) Find the particular solution $y = f(x)$ to the given differential equation with the initial condition $f(-1) = 2$.