Consider the differential equation $\frac { d y } { d x } = \frac { 1 } { 3 } x ( y - 2 ) ^ { 2 }$. (a) A slope field for the given differential equation is shown below. Sketch the solution curve that passes through the point $( 0,2 )$, and sketch the solution curve that passes through the point $( 1,0 )$. (b) Let $y = f ( x )$ be the particular solution to the given differential equation with initial condition $f ( 1 ) = 0$. Write an equation for the line tangent to the graph of $y = f ( x )$ at $x = 1$. Use your equation to approximate $f ( 0.7 )$. (c) Find the particular solution $y = f ( x )$ to the given differential equation with initial condition $f ( 1 ) = 0$.
Consider the differential equation $\frac { d y } { d x } = \frac { 1 } { 3 } x ( y - 2 ) ^ { 2 }$.
(a) A slope field for the given differential equation is shown below. Sketch the solution curve that passes through the point $( 0,2 )$, and sketch the solution curve that passes through the point $( 1,0 )$.
(b) Let $y = f ( x )$ be the particular solution to the given differential equation with initial condition $f ( 1 ) = 0$. Write an equation for the line tangent to the graph of $y = f ( x )$ at $x = 1$. Use your equation to approximate $f ( 0.7 )$.
(c) Find the particular solution $y = f ( x )$ to the given differential equation with initial condition $f ( 1 ) = 0$.