Consider the differential equation $\frac { d y } { d x } = y ^ { 2 } ( 2 x + 2 )$. Let $y = f ( x )$ be the particular solution to the differential equation with initial condition $f ( 0 ) = - 1$. (a) Find $\lim _ { x \rightarrow 0 } \frac { f ( x ) + 1 } { \sin x }$. Show the work that leads to your answer. (b) Use Euler's method, starting at $x = 0$ with two steps of equal size, to approximate $f \left( \frac { 1 } { 2 } \right)$. (c) Find $y = f ( x )$, the particular solution to the differential equation with initial condition $f ( 0 ) = - 1$.
Consider the differential equation $\frac { d y } { d x } = y ^ { 2 } ( 2 x + 2 )$. Let $y = f ( x )$ be the particular solution to the differential equation with initial condition $f ( 0 ) = - 1$.
(a) Find $\lim _ { x \rightarrow 0 } \frac { f ( x ) + 1 } { \sin x }$. Show the work that leads to your answer.
(b) Use Euler's method, starting at $x = 0$ with two steps of equal size, to approximate $f \left( \frac { 1 } { 2 } \right)$.
(c) Find $y = f ( x )$, the particular solution to the differential equation with initial condition $f ( 0 ) = - 1$.